Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001754
(load all 2 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001754: Lattices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
 => ([(0,1)],2)
 => 2
[1,2] => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 5
[2,1] => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 14
[1,3,2] => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 13
[2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 13
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 42
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 42
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 48
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 42
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 132
Description
The number of tolerances of a finite lattice. Let $L$ be a lattice. A tolerance $\tau$ is a reflexive and symmetric relation on $L$ which is compatible with meet and join. Equivalently, a tolerance of $L$ is the image of a congruence by a surjective lattice homomorphism onto $L$. The number of tolerances of a chain of $n$ elements is the Catalan number $\frac{1}{n+1}\binom{2n}{n}$, see [2].
Matching statistic: St000072
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000072: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 38%
Values
[1] => [1,0]
 => [[1],[2]]
 => [[1,1],[1]]
 => 1 = 2 - 1
[1,2] => [1,0,1,0]
 => [[1,3],[2,4]]
 => [[2,2,0,0],[2,1,0],[1,1],[1]]
 => 4 = 5 - 1
[2,1] => [1,1,0,0]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 3 = 4 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 14 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => ? = 13 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 13 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 42 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 42 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => ? = 48 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 42 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 132 - 1
Description
The number of circled entries. asdasda An entry of a Gelfand-Tsetlin pattern is circled if $a_{i,j} = a_{i-1,j}$ (the northeast neighbor is the same).
Matching statistic: St000073
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000073: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 38%
Values
[1] => [1,0]
 => [[1],[2]]
 => [[1,1],[1]]
 => 1 = 2 - 1
[1,2] => [1,0,1,0]
 => [[1,3],[2,4]]
 => [[2,2,0,0],[2,1,0],[1,1],[1]]
 => 4 = 5 - 1
[2,1] => [1,1,0,0]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 3 = 4 - 1
[1,2,3] => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 14 - 1
[1,3,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => ? = 13 - 1
[2,1,3] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 13 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 42 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[4,2,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 42 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => ? = 48 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => ? = 42 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [[5,5,0,0,0,0,0,0,0,0],[5,4,0,0,0,0,0,0,0],[4,4,0,0,0,0,0,0],[4,3,0,0,0,0,0],[3,3,0,0,0,0],[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => ? = 132 - 1
Description
The number of boxed entries. An entry of a Gelfand-Tsetlin pattern is boxed if $a_{i,j} = a_{i-1,j-1}$ (the northwest neighbor is the same).
Matching statistic: St001583
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 38%
Values
[1] => [1,0]
 => [[1],[2]]
 => [2,1] => 0 = 2 - 2
[1,2] => [1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => 3 = 5 - 2
[2,1] => [1,1,0,0]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 2 = 4 - 2
[1,2,3] => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => ? = 14 - 2
[1,3,2] => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => ? = 13 - 2
[2,1,3] => [1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => ? = 13 - 2
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => ? = 42 - 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => ? = 42 - 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => ? = 48 - 2
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => ? = 42 - 2
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [2,4,6,8,10,1,3,5,7,9] => ? = 132 - 2
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001875
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 25% ●values known / values provided: 27%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => ([],1)
 => ([],1)
 => ? = 2 - 10
[1,2] => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? = 5 - 10
[2,1] => [1,2] => ([],2)
 => ([],1)
 => ? = 4 - 10
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 4 = 14 - 10
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 13 - 10
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 3 = 13 - 10
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 42 - 10
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 42 - 10
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 48 - 10
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 42 - 10
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
 => ? = 132 - 10
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000455
Mapping
Mp00209: Permutations —pattern poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 27%●distinct values known / distinct values provided: 25%
Values
[1] => ([],1)
 => ([],1)
 => ([],1)
 => ? = 2 - 14
[1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 5 - 14
[2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],2)
 => ? = 4 - 14
[1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 0 = 14 - 14
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => -1 = 13 - 14
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => -1 = 13 - 14
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 42 - 14
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ? = 42 - 14
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
 => ? = 48 - 14
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => ? = 42 - 14
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 132 - 14
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.