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Your data matches 475 different statistics following compositions of up to 3 maps.
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Matching statistic: St000344
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Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000344: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000344: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => ([(0,1)],2)
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> 0
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> 0
[2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 0
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 0
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[4,2,3,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 0
Description
The number of strongly connected outdegree sequences of a graph.
This is the evaluation of the Tutte polynomial at $x=0$ and $y=1$. According to [1,2], the set of strongly connected outdegree sequences is in bijection with strongly connected minimal orientations and also with external spanning trees.
Matching statistic: St000491
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The number of inversions of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000496
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000497
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The lcb statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000555
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000555: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000555: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,3},{2}} in a set partition.
Matching statistic: St000561
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000561: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000561: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
[3,1,2] => [2,3,1] => {{1,2,3}}
=> 1
[3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 0
[1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
[3,1,2,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 1
[3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
[3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
[3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
[4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 0
[1,2,5,3,4] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 0
[1,4,2,3,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,4,3,5,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 0
[1,4,5,2,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 0
[1,5,3,4,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 0
[2,4,5,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 0
[3,1,2,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 1
[3,2,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 0
[3,2,4,1,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 0
[3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 0
[3,4,1,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,2,3}} in a set partition.
Matching statistic: St000572
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000572: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000572: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The dimension exponent of a set partition.
This is
$$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$
where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$.
It is thus equal to the difference [[St000728]] - [[St000211]].
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block.
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
Matching statistic: St000581
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000581: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal.
Matching statistic: St000582
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000582: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000582: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block.
Matching statistic: St000585
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000585: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000585: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => {{1,2}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 0
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[2,3,1] => [3,1,2] => {{1,2,3}}
=> 0
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 1
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[1,3,4,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[2,3,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[2,3,4,1] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[3,2,4,1] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,4,1,2] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[4,2,3,1] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[1,2,5,3,4] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,3,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[1,4,3,5,2] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[1,4,5,2,3] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[1,5,3,4,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[2,1,4,5,3] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[2,3,1,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 0
[2,3,1,5,4] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[2,3,4,1,5] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[2,3,4,5,1] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 0
[2,4,5,1,3] => [4,1,2,5,3] => {{1,2,3,4,5}}
=> 0
[3,1,2,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[3,2,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[3,2,4,1,5] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[3,2,4,5,1] => [2,5,1,3,4] => {{1,2,3,4,5}}
=> 0
[3,4,1,2,5] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block.
The following 465 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St001172The number of 1-rises at odd height of a Dyck path. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001736The total number of cycles in a graph. St001781The interlacing number of a set partition. St001797The number of overfull subgraphs of a graph. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000272The treewidth of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000392The length of the longest run of ones in a binary word. St000536The pathwidth of a graph. St000729The minimal arc length of a set partition. St000730The maximal arc length of a set partition. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000010The length of the partition. St000147The largest part of an integer partition. St000668The least common multiple of the parts of the partition. St001029The size of the core of a graph. St001062The maximal size of a block of a set partition. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000297The number of leading ones in a binary word. St000347The inversion sum of a binary word. St000387The matching number of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000535The rank-width of a graph. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000944The 3-degree of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001071The beta invariant of the graph. St001091The number of parts in an integer partition whose next smaller part has the same size. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001280The number of parts of an integer partition that are at least two. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001485The modular major index of a binary word. St001512The minimum rank of a graph. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001638The book thickness of a graph. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001961The sum of the greatest common divisors of all pairs of parts. St000048The multinomial of the parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000182The number of permutations whose cycle type is the given integer partition. St000268The number of strongly connected orientations of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000442The maximal area to the right of an up step of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000920The logarithmic height of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001073The number of nowhere zero 3-flows of a graph. St001093The detour number of a graph. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001261The Castelnuovo-Mumford regularity of a graph. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001674The number of vertices of the largest induced star graph in the graph. St001716The 1-improper chromatic number of a graph. St001732The number of peaks visible from the left. St001777The number of weak descents in an integer composition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001933The largest multiplicity of a part in an integer partition. St000013The height of a Dyck path. St000105The number of blocks in the set partition. St000288The number of ones in a binary word. St000381The largest part of an integer composition. St000444The length of the maximal rise of a Dyck path. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000095The number of triangles of a graph. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St000934The 2-degree of an integer partition. St001330The hat guessing number of a graph. St000822The Hadwiger number of the graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000093The cardinality of a maximal independent set of vertices of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000264The girth of a graph, which is not a tree. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001718The number of non-empty open intervals in a poset. St000528The height of a poset. St001343The dimension of the reduced incidence algebra of a poset. St001717The largest size of an interval in a poset. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000648The number of 2-excedences of a permutation. St000662The staircase size of the code of a permutation. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000647The number of big descents of a permutation. St001394The genus of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001845The number of join irreducibles minus the rank of a lattice. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000058The order of a permutation. St000731The number of double exceedences of a permutation. St000223The number of nestings in the permutation. St000358The number of occurrences of the pattern 31-2. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000732The number of double deficiencies of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000886The number of permutations with the same antidiagonal sums. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000485The length of the longest cycle of a permutation. St000527The width of the poset. St000360The number of occurrences of the pattern 32-1. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001728The number of invisible descents of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000546The number of global descents of a permutation. St000422The energy of a graph, if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000989The number of final rises of a permutation. St000779The tier of a permutation. St000094The depth of an ordered tree. St000455The second largest eigenvalue of a graph if it is integral. St000990The first ascent of a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St001727The number of invisible inversions of a permutation. St001396Number of triples of incomparable elements in a finite poset. St000260The radius of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000141The maximum drop size of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001175The size of a partition minus the hook length of the base cell. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000618The number of self-evacuating tableaux of given shape. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000534The number of 2-rises of a permutation. St000355The number of occurrences of the pattern 21-3. St000359The number of occurrences of the pattern 23-1. St000028The number of stack-sorts needed to sort a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St001083The number of boxed occurrences of 132 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000218The number of occurrences of the pattern 213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St000002The number of occurrences of the pattern 123 in a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000078The number of alternating sign matrices whose left key is the permutation. St000365The number of double ascents of a permutation. St000663The number of right floats of a permutation. St001130The number of two successive successions in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000308The height of the tree associated to a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000741The Colin de Verdière graph invariant. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000654The first descent of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000317The cycle descent number of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000039The number of crossings of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000357The number of occurrences of the pattern 12-3. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000379The number of Hamiltonian cycles in a graph. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001309The number of four-cliques in a graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St000021The number of descents of a permutation. St000080The rank of the poset. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000362The size of a minimal vertex cover of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000537The cutwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000948The chromatic discriminant of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001270The bandwidth of a graph. St001281The normalized isoperimetric number of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St000087The number of induced subgraphs. St000166The depth minus 1 of an ordered tree. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001530The depth of a Dyck path. St001581The achromatic number of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001670The connected partition number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St001706The number of closed sets in a graph. St000219The number of occurrences of the pattern 231 in a permutation. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001626The number of maximal proper sublattices of a lattice. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001964The interval resolution global dimension of a poset. St001875The number of simple modules with projective dimension at most 1. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001061The number of indices that are both descents and recoils of a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001811The Castelnuovo-Mumford regularity of a permutation. St001871The number of triconnected components of a graph. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001866The nesting alignments of a signed permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001851The number of Hecke atoms of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001867The number of alignments of type EN of a signed permutation.
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