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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000177
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Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00078: Gelfand-Tsetlin patterns —Schuetzenberger involution⟶ Gelfand-Tsetlin patterns
St000177: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00078: Gelfand-Tsetlin patterns —Schuetzenberger involution⟶ Gelfand-Tsetlin patterns
St000177: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,0],[1]]
=> [[1]]
=> [[1]]
=> [[1]]
=> 0
[[2,0],[0]]
=> [[2,2]]
=> [[2,0],[0]]
=> [[2,0],[2]]
=> 0
[[2,0],[1]]
=> [[1,2]]
=> [[2,0],[1]]
=> [[2,0],[1]]
=> 0
[[1,1],[1]]
=> [[1],[2]]
=> [[1,1],[1]]
=> [[1,1],[1]]
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> [[1]]
=> [[1]]
=> 0
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> [[2,0],[0]]
=> [[2,0],[2]]
=> 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> [[2,0],[1]]
=> [[2,0],[1]]
=> 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [[1,1],[1]]
=> [[1,1],[1]]
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [[1]]
=> [[1]]
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [[3,0,0],[0,0],[0]]
=> [[3,0,0],[3,0],[3]]
=> 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [[3,0,0],[1,0],[0]]
=> [[3,0,0],[3,0],[2]]
=> 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [[3,0,0],[1,0],[1]]
=> [[3,0,0],[2,0],[2]]
=> 0
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [[3,0,0],[2,0],[0]]
=> [[3,0,0],[3,0],[1]]
=> 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> [[3,0,0],[2,0],[1]]
=> 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [[3,0,0],[2,0],[2]]
=> [[3,0,0],[1,0],[1]]
=> 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [[2,1,0],[1,0],[0]]
=> [[2,1,0],[2,1],[2]]
=> 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [[2,1,0],[1,0],[1]]
=> [[2,1,0],[2,0],[2]]
=> 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> [[2,1,0],[2,0],[1]]
=> 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [[2,1,0],[2,0],[0]]
=> [[2,1,0],[2,1],[1]]
=> 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[2,1,0],[1,1],[1]]
=> 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [[2,1,0],[2,0],[2]]
=> [[2,1,0],[1,0],[1]]
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> [[1,1,1],[1,1],[1]]
=> 0
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> [[2,0],[0]]
=> [[2,0],[2]]
=> 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> [[2,0],[1]]
=> [[2,0],[1]]
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [[1,1],[1]]
=> [[1,1],[1]]
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [[1]]
=> [[1]]
=> 0
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> [[3,0,0],[0,0],[0]]
=> [[3,0,0],[3,0],[3]]
=> 0
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> [[3,0,0],[1,0],[0]]
=> [[3,0,0],[3,0],[2]]
=> 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> [[3,0,0],[1,0],[1]]
=> [[3,0,0],[2,0],[2]]
=> 0
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> [[3,0,0],[2,0],[0]]
=> [[3,0,0],[3,0],[1]]
=> 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> [[3,0,0],[2,0],[1]]
=> 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> [[3,0,0],[2,0],[2]]
=> [[3,0,0],[1,0],[1]]
=> 0
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [[2,1,0],[1,0],[0]]
=> [[2,1,0],[2,1],[2]]
=> 0
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [[2,1,0],[1,0],[1]]
=> [[2,1,0],[2,0],[2]]
=> 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> [[2,1,0],[2,0],[1]]
=> 0
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [[2,1,0],[2,0],[0]]
=> [[2,1,0],[2,1],[1]]
=> 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[2,1,0],[1,1],[1]]
=> 0
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [[2,1,0],[2,0],[2]]
=> [[2,1,0],[1,0],[1]]
=> 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> [[1,1,1],[1,1],[1]]
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> [[2,0],[0]]
=> [[2,0],[2]]
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> [[2,0],[1]]
=> [[2,0],[1]]
=> 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [[1,1],[1]]
=> [[1,1],[1]]
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> [[1]]
=> [[1]]
=> 0
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> [[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,0,0,0],[4,0,0],[4,0],[4]]
=> 0
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> [[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[4,0,0,0],[4,0,0],[4,0],[3]]
=> 0
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> [[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[4,0,0,0],[4,0,0],[3,0],[3]]
=> 0
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> [[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[4,0,0,0],[3,0,0],[3,0],[3]]
=> 0
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> [[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[4,0,0,0],[4,0,0],[4,0],[2]]
=> 0
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> [[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[4,0,0,0],[4,0,0],[3,0],[2]]
=> 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> [[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[4,0,0,0],[3,0,0],[3,0],[2]]
=> 1
Description
The number of free tiles in the pattern.
The ''tiling'' of a pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called ''tiles'', and each entry in a pattern belong to exactly one tile.
A tile is ''free'' if it does not intersect any of the first and the last row.
Matching statistic: St000259
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 0
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 0
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000302
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000302: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 0
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
Description
The determinant of the distance matrix of a connected graph.
Matching statistic: St000466
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000466: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 0
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
Description
The Gutman (or modified Schultz) index of a connected graph.
This is
$$\sum_{\{u,v\}\subseteq V} d(u)d(v)d(u,v)$$
where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$.
For trees on $n$ vertices, the modified Schultz index is related to the Wiener index via $S^\ast(T)=4W(T)-(n-1)(2n-1)$ [1].
Matching statistic: St000467
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000467: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 0
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 0
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 0
Description
The hyper-Wiener index of a connected graph.
This is
$$
\sum_{\{u,v\}\subseteq V} d(u,v)+d(u,v)^2.
$$
Matching statistic: St000771
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2 + 1
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $2$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St000772
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2 + 1
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000777
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2 + 1
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001645
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 20%
Values
[[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[3,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ? = 0 + 1
[[3,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ? = 1 + 1
[[3,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,1,0,0],[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ? = 0 + 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[4,0,0,0],[0,0,0],[0,0],[0]]
=> [[4,4,4,4]]
=> ([(0,2),(2,6),(2,7),(3,25),(4,28),(5,24),(6,9),(6,29),(7,10),(7,29),(8,27),(9,8),(9,30),(10,26),(11,17),(11,23),(12,18),(12,22),(13,31),(14,34),(15,1),(16,5),(16,34),(17,3),(17,33),(18,4),(18,32),(19,15),(20,17),(20,31),(21,13),(21,32),(22,11),(22,20),(22,32),(23,16),(23,33),(24,15),(25,19),(26,18),(26,21),(27,13),(27,20),(28,14),(28,16),(29,12),(29,26),(29,30),(30,21),(30,22),(30,27),(31,14),(31,33),(32,23),(32,28),(32,31),(33,25),(33,34),(34,19),(34,24)],35)
=> ([(4,9),(4,27),(5,8),(5,26),(6,12),(6,26),(6,33),(7,13),(7,27),(7,34),(8,12),(8,33),(9,13),(9,34),(10,17),(10,21),(10,23),(10,26),(10,32),(10,33),(11,18),(11,22),(11,24),(11,27),(11,32),(11,34),(12,17),(12,21),(12,23),(12,26),(12,32),(13,18),(13,22),(13,24),(13,27),(13,32),(14,25),(14,28),(14,29),(14,30),(14,31),(14,32),(14,33),(14,34),(15,19),(15,21),(15,23),(15,26),(15,29),(15,30),(15,32),(15,33),(16,20),(16,22),(16,24),(16,27),(16,28),(16,31),(16,32),(16,34),(17,19),(17,21),(17,26),(17,29),(17,30),(17,33),(18,20),(18,22),(18,27),(18,28),(18,31),(18,34),(19,23),(19,25),(19,28),(19,30),(19,31),(19,32),(19,33),(19,34),(20,24),(20,25),(20,29),(20,30),(20,31),(20,32),(20,33),(20,34),(21,25),(21,28),(21,30),(21,31),(21,33),(21,34),(22,25),(22,29),(22,30),(22,31),(22,33),(22,34),(23,25),(23,28),(23,29),(23,30),(23,31),(23,33),(23,34),(24,25),(24,28),(24,29),(24,30),(24,31),(24,33),(24,34),(25,26),(25,27),(25,28),(25,29),(25,32),(26,28),(26,30),(26,31),(26,33),(26,34),(27,29),(27,30),(27,31),(27,33),(27,34),(28,29),(28,30),(28,32),(28,33),(29,31),(29,32),(29,34),(30,32),(31,32),(32,33),(32,34)],35)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[0,0],[0]]
=> [[3,4,4,4]]
=> ([(0,1),(1,5),(1,6),(2,24),(3,27),(4,23),(5,8),(5,28),(6,9),(6,28),(7,26),(8,7),(8,29),(9,25),(10,16),(10,22),(11,17),(11,21),(13,30),(14,33),(15,4),(15,33),(16,2),(16,32),(17,3),(17,31),(18,16),(18,30),(19,12),(20,13),(20,31),(21,10),(21,18),(21,31),(22,15),(22,32),(23,12),(24,19),(25,17),(25,20),(26,13),(26,18),(27,14),(27,15),(28,11),(28,25),(28,29),(29,20),(29,21),(29,26),(30,14),(30,32),(31,22),(31,27),(31,30),(32,24),(32,33),(33,19),(33,23)],34)
=> ([(3,8),(3,26),(4,7),(4,25),(5,11),(5,25),(5,32),(6,12),(6,26),(6,33),(7,11),(7,32),(8,12),(8,33),(9,16),(9,20),(9,22),(9,25),(9,31),(9,32),(10,17),(10,21),(10,23),(10,26),(10,31),(10,33),(11,16),(11,20),(11,22),(11,25),(11,31),(12,17),(12,21),(12,23),(12,26),(12,31),(13,24),(13,27),(13,28),(13,29),(13,30),(13,31),(13,32),(13,33),(14,18),(14,20),(14,22),(14,25),(14,28),(14,29),(14,31),(14,32),(15,19),(15,21),(15,23),(15,26),(15,27),(15,30),(15,31),(15,33),(16,18),(16,20),(16,25),(16,28),(16,29),(16,32),(17,19),(17,21),(17,26),(17,27),(17,30),(17,33),(18,22),(18,24),(18,27),(18,29),(18,30),(18,31),(18,32),(18,33),(19,23),(19,24),(19,28),(19,29),(19,30),(19,31),(19,32),(19,33),(20,24),(20,27),(20,29),(20,30),(20,32),(20,33),(21,24),(21,28),(21,29),(21,30),(21,32),(21,33),(22,24),(22,27),(22,28),(22,29),(22,30),(22,32),(22,33),(23,24),(23,27),(23,28),(23,29),(23,30),(23,32),(23,33),(24,25),(24,26),(24,27),(24,28),(24,31),(25,27),(25,29),(25,30),(25,32),(25,33),(26,28),(26,29),(26,30),(26,32),(26,33),(27,28),(27,29),(27,31),(27,32),(28,30),(28,31),(28,33),(29,31),(30,31),(31,32),(31,33)],34)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[0]]
=> [[2,4,4,4]]
=> ([(0,1),(1,4),(1,5),(2,23),(3,16),(4,7),(4,24),(5,6),(5,24),(6,22),(7,8),(7,25),(8,21),(9,13),(9,20),(10,15),(10,19),(11,28),(12,29),(13,26),(14,3),(14,28),(15,2),(15,27),(17,13),(17,29),(18,12),(18,27),(19,9),(19,17),(19,27),(20,14),(20,26),(21,12),(21,17),(22,15),(22,18),(23,11),(23,14),(24,10),(24,22),(24,25),(25,18),(25,19),(25,21),(26,28),(27,20),(27,23),(27,29),(28,16),(29,11),(29,26)],30)
=> ([(3,20),(4,6),(4,22),(5,12),(5,22),(5,29),(6,12),(6,29),(7,9),(7,15),(7,19),(7,20),(7,28),(8,14),(8,18),(8,21),(8,22),(8,28),(8,29),(9,15),(9,17),(9,20),(9,26),(9,27),(10,15),(10,17),(10,19),(10,20),(10,26),(10,27),(10,28),(11,23),(11,24),(11,25),(11,26),(11,27),(11,28),(11,29),(12,14),(12,18),(12,21),(12,22),(12,28),(13,16),(13,18),(13,21),(13,22),(13,24),(13,25),(13,28),(13,29),(14,16),(14,18),(14,22),(14,24),(14,25),(14,29),(15,23),(15,24),(15,25),(15,26),(15,29),(16,21),(16,23),(16,25),(16,26),(16,27),(16,28),(16,29),(17,19),(17,23),(17,24),(17,25),(17,26),(17,28),(17,29),(18,23),(18,25),(18,26),(18,27),(18,29),(19,23),(19,24),(19,25),(19,26),(19,27),(19,29),(20,23),(20,24),(20,25),(20,26),(20,29),(21,23),(21,24),(21,25),(21,26),(21,27),(21,29),(22,23),(22,25),(22,26),(22,27),(22,29),(23,24),(23,27),(23,28),(24,26),(24,27),(24,28),(25,27),(25,28),(26,28),(27,28),(27,29),(28,29)],30)
=> ? = 0 + 1
[[4,0,0,0],[1,0,0],[1,0],[1]]
=> [[1,4,4,4]]
=> ([(0,9),(2,16),(2,17),(3,13),(4,12),(5,10),(6,11),(7,5),(7,15),(8,6),(8,15),(9,7),(9,8),(10,14),(10,16),(11,14),(11,17),(12,18),(13,18),(14,19),(15,2),(15,10),(15,11),(16,4),(16,19),(17,3),(17,19),(18,1),(19,12),(19,13)],20)
=> ([(4,5),(4,17),(5,16),(6,7),(6,18),(7,19),(8,18),(8,19),(9,16),(9,17),(10,13),(10,14),(10,15),(10,16),(10,17),(11,12),(11,14),(11,15),(11,16),(11,19),(12,13),(12,15),(12,17),(12,18),(13,14),(13,16),(13,19),(14,17),(14,18),(15,18),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[0,0],[0]]
=> [[3,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,24),(3,21),(4,7),(4,25),(5,6),(5,25),(6,23),(7,8),(7,26),(8,22),(9,16),(9,20),(10,15),(10,19),(11,29),(12,29),(14,30),(15,2),(15,28),(16,3),(16,27),(17,16),(17,30),(18,14),(18,28),(19,9),(19,17),(19,28),(20,12),(20,27),(21,13),(22,14),(22,17),(23,15),(23,18),(24,11),(24,12),(25,10),(25,23),(25,26),(26,18),(26,19),(26,22),(27,21),(27,29),(28,20),(28,24),(28,30),(29,13),(30,11),(30,27)],31)
=> ([(3,6),(3,26),(4,8),(4,28),(5,12),(5,26),(5,29),(6,12),(6,29),(7,11),(7,19),(7,22),(7,28),(7,30),(8,11),(8,19),(8,22),(8,30),(9,15),(9,18),(9,21),(9,26),(9,29),(9,30),(10,17),(10,19),(10,22),(10,25),(10,27),(10,28),(10,30),(11,17),(11,19),(11,25),(11,27),(11,28),(12,15),(12,18),(12,21),(12,26),(12,30),(13,16),(13,18),(13,21),(13,23),(13,24),(13,26),(13,29),(13,30),(14,20),(14,23),(14,24),(14,25),(14,27),(14,28),(14,29),(14,30),(15,16),(15,18),(15,23),(15,24),(15,26),(15,29),(16,20),(16,21),(16,23),(16,25),(16,27),(16,28),(16,29),(16,30),(17,20),(17,22),(17,23),(17,24),(17,25),(17,28),(17,29),(17,30),(18,20),(18,23),(18,25),(18,27),(18,28),(18,29),(19,20),(19,23),(19,24),(19,25),(19,28),(19,29),(20,21),(20,22),(20,24),(20,26),(20,27),(20,30),(21,23),(21,24),(21,25),(21,27),(21,28),(21,29),(22,23),(22,24),(22,25),(22,27),(22,28),(22,29),(23,26),(23,27),(23,30),(24,25),(24,27),(24,28),(24,30),(25,26),(25,30),(26,27),(26,28),(26,29),(27,29),(27,30),(28,30),(29,30)],31)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[1,0],[0]]
=> [[2,3,4,4]]
=> ([(0,1),(1,3),(1,4),(2,21),(3,6),(3,22),(4,5),(4,22),(5,20),(6,7),(6,23),(7,19),(8,13),(8,18),(9,14),(9,17),(10,26),(11,26),(12,27),(13,24),(14,2),(14,25),(15,13),(15,27),(16,12),(16,25),(17,8),(17,15),(17,25),(18,10),(18,24),(19,12),(19,15),(20,14),(20,16),(21,10),(21,11),(22,9),(22,20),(22,23),(23,16),(23,17),(23,19),(24,26),(25,18),(25,21),(25,27),(27,11),(27,24)],28)
=> ([(3,5),(3,22),(4,11),(4,22),(4,27),(5,11),(5,27),(6,7),(6,14),(6,18),(6,26),(7,14),(7,17),(7,24),(7,25),(8,14),(8,17),(8,18),(8,24),(8,25),(8,26),(9,13),(9,16),(9,19),(9,22),(9,26),(9,27),(10,20),(10,21),(10,23),(10,24),(10,25),(10,26),(10,27),(11,13),(11,16),(11,19),(11,22),(11,26),(12,15),(12,16),(12,19),(12,21),(12,22),(12,23),(12,26),(12,27),(13,15),(13,16),(13,21),(13,22),(13,23),(13,27),(14,20),(14,21),(14,23),(14,24),(14,27),(15,19),(15,20),(15,23),(15,24),(15,25),(15,26),(15,27),(16,20),(16,23),(16,24),(16,25),(16,27),(17,18),(17,20),(17,21),(17,23),(17,24),(17,26),(17,27),(18,20),(18,21),(18,23),(18,24),(18,25),(18,27),(19,20),(19,21),(19,23),(19,24),(19,25),(19,27),(20,21),(20,22),(20,25),(20,26),(21,24),(21,25),(21,26),(22,23),(22,24),(22,25),(22,27),(23,25),(23,26),(24,26),(25,26),(25,27),(26,27)],28)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[1,0],[1]]
=> [[1,3,4,4]]
=> ([(0,1),(1,4),(1,5),(2,14),(3,13),(4,6),(4,17),(5,7),(5,17),(6,15),(7,16),(8,11),(8,12),(10,18),(11,3),(11,18),(12,2),(12,18),(13,9),(14,9),(15,10),(15,11),(16,10),(16,12),(17,8),(17,15),(17,16),(18,13),(18,14)],19)
=> ([(3,4),(3,16),(4,15),(5,6),(5,17),(6,18),(7,17),(7,18),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,13),(10,14),(10,15),(10,18),(11,12),(11,14),(11,16),(11,17),(12,13),(12,15),(12,18),(13,16),(13,17),(14,17),(14,18),(15,16),(15,17),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[0]]
=> [[2,2,4,4]]
=> ([(0,1),(1,3),(1,4),(2,15),(3,6),(3,18),(4,5),(4,18),(5,17),(6,7),(6,19),(7,16),(8,12),(8,14),(10,21),(11,21),(12,2),(12,20),(13,11),(13,20),(14,10),(14,20),(15,9),(16,10),(16,11),(17,12),(17,13),(18,8),(18,17),(18,19),(19,13),(19,14),(19,16),(20,15),(20,21),(21,9)],22)
=> ([(3,6),(3,19),(4,8),(4,20),(5,13),(5,19),(5,21),(6,13),(6,21),(7,12),(7,17),(7,18),(7,20),(7,21),(8,12),(8,17),(8,18),(8,21),(9,11),(9,15),(9,16),(9,19),(9,20),(9,21),(10,14),(10,15),(10,16),(10,17),(10,18),(10,20),(10,21),(11,13),(11,14),(11,16),(11,17),(11,18),(11,21),(12,14),(12,15),(12,16),(12,17),(12,20),(13,15),(13,16),(13,19),(13,20),(14,15),(14,18),(14,19),(14,20),(14,21),(15,17),(15,18),(15,21),(16,18),(16,19),(16,21),(17,19),(17,20),(18,19),(18,20),(19,21),(20,21)],22)
=> ? = 0 + 1
[[4,0,0,0],[2,0,0],[2,0],[1]]
=> [[1,2,4,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[2,0,0],[2,0],[2]]
=> [[1,1,4,4]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[0,0],[0]]
=> [[3,3,3,4]]
=> ([(0,1),(1,3),(1,4),(2,14),(3,6),(3,20),(4,5),(4,20),(5,19),(6,7),(6,21),(7,18),(8,12),(8,13),(9,11),(9,17),(10,22),(11,24),(12,23),(13,2),(13,23),(15,13),(15,22),(16,10),(16,24),(17,8),(17,15),(17,24),(18,10),(18,15),(19,11),(19,16),(20,9),(20,19),(20,21),(21,16),(21,17),(21,18),(22,23),(23,14),(24,12),(24,22)],25)
=> ([(3,19),(4,6),(4,24),(5,9),(5,23),(5,24),(6,9),(6,23),(7,12),(7,15),(7,19),(7,21),(8,14),(8,20),(8,22),(8,23),(8,24),(9,14),(9,20),(9,22),(9,24),(10,16),(10,17),(10,18),(10,20),(10,22),(10,23),(10,24),(11,13),(11,15),(11,17),(11,18),(11,19),(11,21),(11,23),(12,13),(12,15),(12,17),(12,18),(12,19),(12,23),(13,16),(13,18),(13,20),(13,21),(13,22),(13,24),(14,16),(14,17),(14,18),(14,20),(14,23),(14,24),(15,16),(15,18),(15,20),(15,22),(15,24),(16,17),(16,19),(16,21),(16,22),(16,23),(17,20),(17,21),(17,22),(17,24),(18,19),(18,21),(18,22),(19,20),(19,22),(19,24),(20,21),(20,23),(21,22),(21,23),(21,24),(22,23),(23,24)],25)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[1,0],[0]]
=> [[2,3,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,16),(3,6),(3,16),(4,18),(5,17),(6,5),(6,19),(7,9),(7,11),(8,10),(8,14),(9,21),(10,22),(11,21),(12,20),(13,12),(13,22),(14,7),(14,15),(14,22),(15,9),(15,20),(16,8),(16,18),(16,19),(17,12),(17,15),(18,10),(18,13),(19,13),(19,14),(19,17),(20,21),(22,11),(22,20)],23)
=> ([(3,6),(3,21),(4,10),(4,16),(4,20),(5,9),(5,21),(5,22),(6,9),(6,22),(7,15),(7,18),(7,19),(7,21),(7,22),(8,13),(8,14),(8,16),(8,17),(8,20),(8,22),(9,15),(9,18),(9,19),(9,21),(10,13),(10,14),(10,16),(10,17),(10,22),(11,12),(11,14),(11,17),(11,18),(11,19),(11,21),(11,22),(12,13),(12,15),(12,16),(12,17),(12,19),(12,20),(12,22),(13,14),(13,18),(13,19),(13,20),(13,21),(14,15),(14,16),(14,19),(14,20),(15,17),(15,18),(15,21),(15,22),(16,18),(16,19),(16,21),(17,18),(17,19),(17,20),(17,21),(18,20),(18,22),(19,20),(19,22),(20,21),(20,22),(21,22)],23)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[1,0],[1]]
=> [[1,3,3,4]]
=> ([(0,7),(1,11),(1,14),(2,10),(3,8),(4,9),(5,3),(5,13),(6,4),(6,13),(7,5),(7,6),(8,12),(8,14),(9,11),(9,12),(11,15),(12,15),(13,1),(13,8),(13,9),(14,2),(14,15),(15,10)],16)
=> ([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[0]]
=> [[2,2,3,4]]
=> ([(0,1),(1,2),(1,3),(2,4),(2,13),(3,6),(3,13),(4,15),(5,14),(6,5),(6,16),(7,10),(7,12),(8,18),(9,18),(10,17),(11,9),(11,17),(12,8),(12,17),(13,7),(13,15),(13,16),(14,8),(14,9),(15,10),(15,11),(16,11),(16,12),(16,14),(17,18)],19)
=> ([(3,5),(3,17),(4,11),(4,17),(4,18),(5,11),(5,18),(6,8),(6,10),(6,14),(6,18),(7,13),(7,15),(7,16),(7,17),(7,18),(8,10),(8,12),(8,15),(8,16),(9,10),(9,12),(9,14),(9,15),(9,16),(9,18),(10,13),(10,16),(10,17),(11,13),(11,15),(11,16),(11,17),(12,13),(12,14),(12,16),(12,17),(12,18),(13,14),(13,15),(13,18),(14,15),(14,16),(14,17),(15,17),(15,18),(16,18),(17,18)],19)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[2,0],[1]]
=> [[1,2,3,4]]
=> ([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> ([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 2 + 1
[[4,0,0,0],[3,0,0],[2,0],[2]]
=> [[1,1,3,4]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ? = 1 + 1
[[4,0,0,0],[3,0,0],[3,0],[0]]
=> [[2,2,2,4]]
=> ([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> ([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> ? = 0 + 1
[[4,0,0,0],[3,0,0],[3,0],[1]]
=> [[1,2,2,4]]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 1 + 1
[[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
=> [[1],[2],[3],[4],[5]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
[[1]]
=> [[1]]
=> ([],1)
=> ([],1)
=> 1 = 0 + 1
Description
The pebbling number of a connected graph.
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