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Your data matches 121 different statistics following compositions of up to 3 maps.
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Matching statistic: St000829
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
St000829: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 2
[3,2,1,4] => 2
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 2
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 2
[4,3,1,2] => 2
[4,3,2,1] => 3
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The Ulam distance of a permutation to the identity permutation.
This is, for a permutation $\pi$ of $n$, given by $n$ minus the length of the longest increasing subsequence of $\pi^{-1}$.
In other words, this statistic plus [[St000062]] equals $n$.
Matching statistic: St000354
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => 1
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 2
[1,3,5,4,2] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => 2
[1,4,5,3,2] => [1,2,5,4,3] => 2
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000362
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00160: Permutations —graph of inversions⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => ([],2)
=> 0
[2,1] => ([(0,1)],2)
=> 1
[1,2,3] => ([],3)
=> 0
[1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => ([(1,3),(2,3)],4)
=> 1
[1,4,2,3] => ([(1,3),(2,3)],4)
=> 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => ([(0,3),(1,2)],4)
=> 2
[2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,2,3,4,5] => ([],5)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> 1
[1,2,4,3,5] => ([(3,4)],5)
=> 1
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,3,2,4,5] => ([(3,4)],5)
=> 1
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 2
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 2
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
Description
The size of a minimal vertex cover of a graph.
A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St001176
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2]
=> 0
[2,1] => [1,1]
=> 1
[1,2,3] => [3]
=> 0
[1,3,2] => [2,1]
=> 1
[2,1,3] => [2,1]
=> 1
[2,3,1] => [2,1]
=> 1
[3,1,2] => [2,1]
=> 1
[3,2,1] => [1,1,1]
=> 2
[1,2,3,4] => [4]
=> 0
[1,2,4,3] => [3,1]
=> 1
[1,3,2,4] => [3,1]
=> 1
[1,3,4,2] => [3,1]
=> 1
[1,4,2,3] => [3,1]
=> 1
[1,4,3,2] => [2,1,1]
=> 2
[2,1,3,4] => [3,1]
=> 1
[2,1,4,3] => [2,2]
=> 2
[2,3,1,4] => [3,1]
=> 1
[2,3,4,1] => [3,1]
=> 1
[2,4,1,3] => [2,2]
=> 2
[2,4,3,1] => [2,1,1]
=> 2
[3,1,2,4] => [3,1]
=> 1
[3,1,4,2] => [2,2]
=> 2
[3,2,1,4] => [2,1,1]
=> 2
[3,2,4,1] => [2,1,1]
=> 2
[3,4,1,2] => [2,2]
=> 2
[3,4,2,1] => [2,1,1]
=> 2
[4,1,2,3] => [3,1]
=> 1
[4,1,3,2] => [2,1,1]
=> 2
[4,2,1,3] => [2,1,1]
=> 2
[4,2,3,1] => [2,1,1]
=> 2
[4,3,1,2] => [2,1,1]
=> 2
[4,3,2,1] => [1,1,1,1]
=> 3
[1,2,3,4,5] => [5]
=> 0
[1,2,3,5,4] => [4,1]
=> 1
[1,2,4,3,5] => [4,1]
=> 1
[1,2,4,5,3] => [4,1]
=> 1
[1,2,5,3,4] => [4,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> 2
[1,3,2,4,5] => [4,1]
=> 1
[1,3,2,5,4] => [3,2]
=> 2
[1,3,4,2,5] => [4,1]
=> 1
[1,3,4,5,2] => [4,1]
=> 1
[1,3,5,2,4] => [3,2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> 2
[1,4,2,3,5] => [4,1]
=> 1
[1,4,2,5,3] => [3,2]
=> 2
[1,4,3,2,5] => [3,1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> 2
[1,4,5,2,3] => [3,2]
=> 2
[1,4,5,3,2] => [3,1,1]
=> 2
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001489
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => 2
[2,4,3,1] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => 1
[3,1,4,2] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => 2
[3,4,2,1] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => 1
[4,1,3,2] => [4,1,3,2] => 2
[4,2,1,3] => [4,2,1,3] => 2
[4,2,3,1] => [4,1,3,2] => 2
[4,3,1,2] => [4,3,1,2] => 2
[4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => 2
[1,3,5,4,2] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => 2
[1,4,5,3,2] => [1,2,5,4,3] => 2
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St000021
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => 2
[3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 1
[4,1,3,2] => [4,1,3,2] => [2,4,3,1] => 2
[4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 2
[4,2,3,1] => [4,1,3,2] => [2,4,3,1] => 2
[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 2
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,4,2,5,3] => 2
[1,3,5,4,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,4,5,2,3] => [1,3,5,2,4] => [1,4,2,5,3] => 2
[1,4,5,3,2] => [1,2,5,4,3] => [1,2,5,4,3] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000157
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2] => [3,1,2] => [[1,2],[3]]
=> 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,3,4,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,4,2,3] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,4,1,3] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[2,4,3,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [3,1,2,4] => [[1,2,4],[3]]
=> 1
[3,1,4,2] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,1,2] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,1,2,3] => [4,1,2,3] => [[1,2,3],[4]]
=> 1
[4,1,3,2] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [4,2,1,3] => [[1,3],[2],[4]]
=> 2
[4,2,3,1] => [4,1,3,2] => [[1,2],[3],[4]]
=> 2
[4,3,1,2] => [4,3,1,2] => [[1,2],[3],[4]]
=> 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,2,4,5,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,3,4,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,3,4,5,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,3,5,4,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,2,5,3] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 2
[1,4,3,5,2] => [1,3,2,5,4] => [[1,2,4],[3,5]]
=> 2
[1,4,5,2,3] => [1,3,5,2,4] => [[1,2,4],[3,5]]
=> 2
[1,4,5,3,2] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000211
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,2]]
=> {{1,2}}
=> 1
[2,1] => [[1],[2]]
=> {{1},{2}}
=> 0
[1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 2
[1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0
[1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2
[2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[2,4,3,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[3,2,4,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
[4,2,1,3] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[4,2,3,1] => [[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> 1
[4,3,1,2] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 3
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 3
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000228
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,1]
=> [1]
=> 1
[2,1] => [2]
=> []
=> 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [2,1]
=> [1]
=> 1
[3,2,1] => [3]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 2
[1,4,3,2] => [3,1]
=> [1]
=> 1
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 2
[2,3,4,1] => [2,1,1]
=> [1,1]
=> 2
[2,4,1,3] => [2,1,1]
=> [1,1]
=> 2
[2,4,3,1] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2]
=> 2
[3,2,1,4] => [3,1]
=> [1]
=> 1
[3,2,4,1] => [3,1]
=> [1]
=> 1
[3,4,1,2] => [2,1,1]
=> [1,1]
=> 2
[3,4,2,1] => [3,1]
=> [1]
=> 1
[4,1,2,3] => [2,1,1]
=> [1,1]
=> 2
[4,1,3,2] => [3,1]
=> [1]
=> 1
[4,2,1,3] => [3,1]
=> [1]
=> 1
[4,2,3,1] => [3,1]
=> [1]
=> 1
[4,3,1,2] => [3,1]
=> [1]
=> 1
[4,3,2,1] => [4]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 3
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 3
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 2
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000245
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [[1,2]]
=> [1,2] => 1
[2,1] => [[1],[2]]
=> [2,1] => 0
[1,2,3] => [[1,2,3]]
=> [1,2,3] => 2
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => 1
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => 1
[2,3,1] => [[1,2],[3]]
=> [3,1,2] => 1
[3,1,2] => [[1,3],[2]]
=> [2,1,3] => 1
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => 0
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => 3
[1,2,4,3] => [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4,1,2,3] => 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,4,1,3] => 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4,1,2,3] => 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,4,1,3] => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[3,2,4,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [4,2,1,3] => 1
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 3
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 3
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1,2,3,5] => 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 3
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => 3
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,5,1,2,4] => 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 3
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 2
Description
The number of ascents of a permutation.
The following 111 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000316The number of non-left-to-right-maxima of a permutation. St000377The dinv defect of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000653The last descent of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000738The first entry in the last row of a standard tableau. St000018The number of inversions of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000051The size of the left subtree of a binary tree. St000074The number of special entries. St000083The number of left oriented leafs of a binary tree except the first one. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000209Maximum difference of elements in cycles. St000216The absolute length of a permutation. St000224The sorting index of a permutation. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000369The dinv deficit of a Dyck path. St000441The number of successions of a permutation. St000502The number of successions of a set partitions. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000728The dimension of a set partition. St000809The reduced reflection length of the permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001034The area of the parallelogram polyomino associated with the Dyck path. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000010The length of the partition. St000054The first entry of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000093The cardinality of a maximal independent set of vertices of a graph. St000213The number of weak exceedances (also weak excedences) of a permutation. St000443The number of long tunnels of a Dyck path. St000702The number of weak deficiencies of a permutation. St000734The last entry in the first row of a standard tableau. St000740The last entry of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000260The radius of a connected graph. St001668The number of points of the poset minus the width of the poset. St001645The pebbling number of a connected graph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001568The smallest positive integer that does not appear twice in the partition. St000741The Colin de Verdière graph invariant. St001896The number of right descents of a signed permutations. 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 $eA$. St001330The hat guessing number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001863The number of weak excedances of a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001712The number of natural descents of a standard Young tableau. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000746The number of pairs with odd minimum in a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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