Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St001629
Mp00317: Integer partitions odd partsBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00133: Integer compositions delta morphismInteger compositions
St001629: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[3,2,1]
=> 101 => [1,1,1] => [3] => 1
[3,2,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[5,2,1]
=> 101 => [1,1,1] => [3] => 1
[3,2,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[3,2,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[5,2,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[4,3,2]
=> 010 => [1,1,1] => [3] => 1
[3,3,2,1]
=> 1101 => [2,1,1] => [1,2] => 0
[3,2,2,1,1]
=> 10011 => [1,2,2] => [1,2] => 0
[3,2,1,1,1,1]
=> 101111 => [1,1,4] => [2,1] => 0
[7,2,1]
=> 101 => [1,1,1] => [3] => 1
[5,4,1]
=> 101 => [1,1,1] => [3] => 1
[5,2,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[5,2,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[4,3,2,1]
=> 0101 => [1,1,1,1] => [4] => 1
[3,3,2,1,1]
=> 11011 => [2,1,2] => [1,1,1] => 1
[3,2,2,2,1]
=> 10001 => [1,3,1] => [1,1,1] => 1
[3,2,2,1,1,1]
=> 100111 => [1,2,3] => [1,1,1] => 1
[3,2,1,1,1,1,1]
=> 1011111 => [1,1,5] => [2,1] => 0
[7,2,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[6,3,2]
=> 010 => [1,1,1] => [3] => 1
[5,4,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[5,3,2,1]
=> 1101 => [2,1,1] => [1,2] => 0
[5,2,2,1,1]
=> 10011 => [1,2,2] => [1,2] => 0
[5,2,1,1,1,1]
=> 101111 => [1,1,4] => [2,1] => 0
[4,3,2,2]
=> 0100 => [1,1,2] => [2,1] => 0
[4,3,2,1,1]
=> 01011 => [1,1,1,2] => [3,1] => 0
[3,3,2,2,1]
=> 11001 => [2,2,1] => [2,1] => 0
[3,3,2,1,1,1]
=> 110111 => [2,1,3] => [1,1,1] => 1
[3,2,2,2,1,1]
=> 100011 => [1,3,2] => [1,1,1] => 1
[3,2,2,1,1,1,1]
=> 1001111 => [1,2,4] => [1,1,1] => 1
[3,2,1,1,1,1,1,1]
=> 10111111 => [1,1,6] => [2,1] => 0
[9,2,1]
=> 101 => [1,1,1] => [3] => 1
[7,4,1]
=> 101 => [1,1,1] => [3] => 1
[7,2,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[7,2,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[6,3,2,1]
=> 0101 => [1,1,1,1] => [4] => 1
[5,4,3]
=> 101 => [1,1,1] => [3] => 1
[5,4,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[5,4,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[5,3,2,1,1]
=> 11011 => [2,1,2] => [1,1,1] => 1
[5,2,2,2,1]
=> 10001 => [1,3,1] => [1,1,1] => 1
[5,2,2,1,1,1]
=> 100111 => [1,2,3] => [1,1,1] => 1
[5,2,1,1,1,1,1]
=> 1011111 => [1,1,5] => [2,1] => 0
[4,3,3,2]
=> 0110 => [1,2,1] => [1,1,1] => 1
[4,3,2,2,1]
=> 01001 => [1,1,2,1] => [2,1,1] => 1
[4,3,2,1,1,1]
=> 010111 => [1,1,1,3] => [3,1] => 0
[3,3,3,2,1]
=> 11101 => [3,1,1] => [1,2] => 0
[3,3,2,2,1,1]
=> 110011 => [2,2,2] => [3] => 1
[3,3,2,1,1,1,1]
=> 1101111 => [2,1,4] => [1,1,1] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000068
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St000068: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
Description
The number of minimal elements in a poset.
Matching statistic: St000908
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St000908: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000914
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St000914: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
Description
The sum of the values of the Möbius function of a poset. The Möbius function $\mu$ of a finite poset is defined as $$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\ -\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\ 0&\text{otherwise}. \end{cases} $$ Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is $$ \sum_{x\leq y} \mu(x,y). $$ If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components. This statistic is also called the magnitude of a poset.
Matching statistic: St001532
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St001532: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
Description
The leading coefficient of the Poincare polynomial of the poset cone. For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$. Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$. This statistic records its leading coefficient.
Matching statistic: St001942
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St001942: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
Description
The number of loops of the quiver corresponding to the reduced incidence algebra of a poset.
Matching statistic: St000281
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St000281: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 - 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 - 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 - 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 - 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 - 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 - 1
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 - 1
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0 - 1
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1 - 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1 - 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1 - 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0 - 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 - 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 - 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 - 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 - 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 - 1
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 - 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0 - 1
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1 - 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1 - 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1 - 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0 - 1
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1 - 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0 - 1
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 - 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
Description
The size of the preimage of the map 'to poset' from Binary trees to Posets.
Matching statistic: St000282
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St000282: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 - 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 - 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 - 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 - 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 - 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 - 1
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 - 1
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0 - 1
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1 - 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1 - 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1 - 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0 - 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 - 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 - 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 - 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 - 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 - 1
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 - 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 - 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0 - 1
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 - 1
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1 - 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1 - 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1 - 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0 - 1
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1 - 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0 - 1
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 - 1
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 - 1
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 - 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0 = 1 - 1
Description
The size of the preimage of the map 'to poset' from Ordered trees to Posets.
Matching statistic: St000298
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St000298: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 + 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 + 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 + 1
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 + 1
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0 + 1
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1 + 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1 + 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1 + 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0 + 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 + 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 + 1
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0 + 1
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1 + 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1 + 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1 + 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0 + 1
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1 + 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0 + 1
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 + 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000632
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St000632: Posets ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[3,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 + 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
[3,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 + 1
[3,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 + 1
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[5,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 + 1
[4,3,2,2]
=> 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[4,3,2,1,1]
=> 01011 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? = 0 + 1
[3,3,2,2,1]
=> 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[3,3,2,1,1,1]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 1 + 1
[3,2,2,2,1,1]
=> 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? = 1 + 1
[3,2,2,1,1,1,1]
=> 1001111 => ([(0,5),(0,6),(1,3),(1,15),(2,14),(3,2),(3,20),(4,8),(4,16),(5,1),(5,13),(5,19),(6,4),(6,13),(6,19),(8,9),(9,10),(10,11),(11,7),(12,7),(13,8),(14,12),(15,18),(15,20),(16,9),(16,18),(17,11),(17,12),(18,10),(18,17),(19,15),(19,16),(20,14),(20,17)],21)
=> ? = 1 + 1
[3,2,1,1,1,1,1,1]
=> 10111111 => ([(0,6),(0,7),(1,3),(1,16),(2,15),(3,4),(3,18),(4,5),(4,20),(5,2),(5,19),(6,14),(6,17),(7,1),(7,14),(7,17),(9,10),(10,11),(11,12),(12,13),(13,8),(14,9),(15,8),(16,10),(16,18),(17,9),(17,16),(18,11),(18,20),(19,13),(19,15),(20,12),(20,19)],21)
=> ? = 0 + 1
[9,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[7,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[6,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[5,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[5,4,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[5,4,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[5,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
[5,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 1 + 1
[5,2,2,1,1,1]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 1 + 1
[5,2,1,1,1,1,1]
=> 1011111 => ([(0,5),(0,6),(1,4),(1,14),(2,13),(3,2),(3,16),(4,3),(4,17),(5,12),(5,15),(6,1),(6,12),(6,15),(8,11),(9,10),(10,8),(11,7),(12,9),(13,7),(14,10),(14,17),(15,9),(15,14),(16,11),(16,13),(17,8),(17,16)],18)
=> ? = 0 + 1
[4,3,3,2]
=> 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[4,3,2,2,1]
=> 01001 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? = 1 + 1
[4,3,2,1,1,1]
=> 010111 => ([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> ? = 0 + 1
[3,3,3,2,1]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 0 + 1
[3,3,2,2,1,1]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 1 + 1
[3,3,2,1,1,1,1]
=> 1101111 => ([(0,4),(0,5),(1,2),(1,17),(2,7),(3,1),(3,8),(3,16),(4,18),(4,19),(5,3),(5,18),(5,19),(7,12),(8,9),(9,10),(10,11),(11,6),(12,6),(13,10),(13,14),(14,11),(14,12),(15,9),(15,13),(16,13),(16,17),(17,7),(17,14),(18,15),(18,16),(19,8),(19,15)],20)
=> ? = 1 + 1
[3,2,2,2,2,1]
=> 100001 => ([(0,4),(0,5),(1,3),(1,7),(1,8),(2,13),(2,14),(3,2),(3,11),(3,12),(4,9),(4,10),(5,1),(5,9),(5,10),(7,12),(8,11),(9,8),(10,7),(11,13),(12,14),(13,6),(14,6)],15)
=> ? = 1 + 1
[3,2,2,2,1,1,1]
=> 1000111 => ([(0,5),(0,6),(1,2),(1,20),(2,8),(3,4),(3,15),(3,21),(4,9),(4,17),(5,1),(5,10),(5,18),(6,3),(6,10),(6,18),(8,11),(9,12),(10,15),(11,13),(12,14),(13,7),(14,7),(15,9),(16,11),(16,19),(17,12),(17,19),(18,20),(18,21),(19,13),(19,14),(20,8),(20,16),(21,16),(21,17)],22)
=> ? = 0 + 1
[3,2,2,1,1,1,1,1]
=> 10011111 => ([(0,6),(0,7),(1,4),(1,17),(2,16),(3,2),(3,23),(4,3),(4,24),(5,9),(5,18),(6,1),(6,15),(6,22),(7,5),(7,15),(7,22),(9,11),(10,12),(11,10),(12,13),(13,8),(14,8),(15,9),(16,14),(17,21),(17,24),(18,11),(18,21),(19,12),(19,20),(20,13),(20,14),(21,10),(21,19),(22,17),(22,18),(23,16),(23,20),(24,19),(24,23)],25)
=> ? = 1 + 1
[3,2,1,1,1,1,1,1,1]
=> 101111111 => ([(0,7),(0,8),(1,3),(1,18),(2,17),(3,5),(3,20),(4,6),(4,22),(5,4),(5,23),(6,2),(6,21),(7,16),(7,19),(8,1),(8,16),(8,19),(10,11),(11,13),(12,14),(13,12),(14,15),(15,9),(16,10),(17,9),(18,11),(18,20),(19,10),(19,18),(20,13),(20,23),(21,15),(21,17),(22,14),(22,21),(23,12),(23,22)],24)
=> ? = 0 + 1
[9,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[8,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[7,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
[7,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 0 + 1
[7,2,1,1,1,1]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 0 + 1
[6,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[11,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[9,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[10,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[8,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[6,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[13,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[11,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[9,6,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[9,4,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[7,6,3]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[12,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[10,5,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[8,7,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
[8,5,4]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 1 + 1
Description
The jump number of the poset. A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001268The size of the largest ordinal summand in the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St001399The distinguishing number of a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001779The order of promotion on the set of linear extensions of a poset. St000080The rank of the poset. St000307The number of rowmotion orbits of a poset. St000526The number of posets with combinatorially isomorphic order polytopes. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000100The number of linear extensions of a poset. St000528The height of a poset. St000633The size of the automorphism group of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000910The number of maximal chains of minimal length in a poset. St000912The number of maximal antichains in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001510The number of self-evacuating linear extensions of a finite poset. St001902The number of potential covers of a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St000189The number of elements in the poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St001717The largest size of an interval in a poset. St000656The number of cuts of a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St001664The number of non-isomorphic subposets of a poset. St000641The number of non-empty boolean intervals in a poset. St000639The number of relations in a poset. St001909The number of interval-closed sets of a poset. St000180The number of chains of a poset. St001709The number of homomorphisms to the three element chain of a poset. St001815The number of order preserving surjections from a poset to a total order. St001813The product of the sizes of the principal order filters in a poset. St000634The number of endomorphisms of a poset.