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Your data matches 529 different statistics following compositions of up to 3 maps.
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Matching statistic: St000118
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000118: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 0
[1,2] => [.,[.,.]]
=> 0
[2,1] => [[.,.],.]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> 0
[2,3,1] => [[.,[.,.]],.]
=> 0
[3,1,2] => [[.,.],[.,.]]
=> 0
[3,2,1] => [[[.,.],.],.]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 0
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 0
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 0
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 0
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 0
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 0
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 0
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 0
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> 1
[3,5,2,4,1] => [[[.,[.,.]],[.,.]],.]
=> 0
[3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> 0
[3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
=> 0
[4,2,5,3,1] => [[[.,.],[[.,.],.]],.]
=> 0
[4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> 0
[4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> 0
[4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
=> 0
[4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> 0
[4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> 0
[4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> 0
[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> 0
[4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> 0
[5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> 0
[5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> 1
[5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> 0
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> 0
[5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> 0
[5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> 0
[5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> 0
[5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> 0
[5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> 0
[5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> 1
[5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> 0
[5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 0
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,.]]]}}} in a binary tree.
[[oeis:A001006]] counts binary trees avoiding this pattern.
Matching statistic: St000752
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0
[1,2] => [2]
=> 0
[2,1] => [1,1]
=> 0
[1,2,3] => [3]
=> 1
[1,3,2] => [2,1]
=> 0
[2,1,3] => [2,1]
=> 0
[2,3,1] => [2,1]
=> 0
[3,1,2] => [2,1]
=> 0
[3,2,1] => [1,1,1]
=> 0
[1,4,3,2] => [2,1,1]
=> 0
[2,3,4,1] => [3,1]
=> 1
[2,4,1,3] => [2,2]
=> 0
[2,4,3,1] => [2,1,1]
=> 0
[3,1,4,2] => [2,2]
=> 0
[3,2,1,4] => [2,1,1]
=> 0
[3,2,4,1] => [2,1,1]
=> 0
[3,4,1,2] => [2,2]
=> 0
[3,4,2,1] => [2,1,1]
=> 0
[4,1,2,3] => [3,1]
=> 1
[4,1,3,2] => [2,1,1]
=> 0
[4,2,1,3] => [2,1,1]
=> 0
[4,2,3,1] => [2,1,1]
=> 0
[4,3,1,2] => [2,1,1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> 1
[3,5,2,4,1] => [2,2,1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> 0
[4,2,5,3,1] => [2,2,1]
=> 0
[4,3,2,5,1] => [2,1,1,1]
=> 0
[4,3,5,1,2] => [2,2,1]
=> 0
[4,3,5,2,1] => [2,1,1,1]
=> 0
[4,5,1,3,2] => [2,2,1]
=> 0
[4,5,2,1,3] => [2,2,1]
=> 0
[4,5,2,3,1] => [2,2,1]
=> 0
[4,5,3,1,2] => [2,2,1]
=> 0
[4,5,3,2,1] => [2,1,1,1]
=> 0
[5,1,4,3,2] => [2,1,1,1]
=> 0
[5,2,3,4,1] => [3,1,1]
=> 1
[5,2,4,1,3] => [2,2,1]
=> 0
[5,2,4,3,1] => [2,1,1,1]
=> 0
[5,3,1,4,2] => [2,2,1]
=> 0
[5,3,2,1,4] => [2,1,1,1]
=> 0
[5,3,2,4,1] => [2,1,1,1]
=> 0
[5,3,4,1,2] => [2,2,1]
=> 0
[5,3,4,2,1] => [2,1,1,1]
=> 0
[5,4,1,2,3] => [3,1,1]
=> 1
[5,4,1,3,2] => [2,1,1,1]
=> 0
[5,4,2,1,3] => [2,1,1,1]
=> 0
Description
The Grundy value for the game 'Couples are forever' on an integer partition.
Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Matching statistic: St001385
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001385: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001385: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 0 + 1
[1,2] => [2]
=> 1 = 0 + 1
[2,1] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> 1 = 0 + 1
[2,3,1] => [2,1]
=> 1 = 0 + 1
[3,1,2] => [2,1]
=> 1 = 0 + 1
[3,2,1] => [1,1,1]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> 1 = 0 + 1
[2,3,4,1] => [3,1]
=> 2 = 1 + 1
[2,4,1,3] => [2,2]
=> 1 = 0 + 1
[2,4,3,1] => [2,1,1]
=> 1 = 0 + 1
[3,1,4,2] => [2,2]
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> 1 = 0 + 1
[3,4,2,1] => [2,1,1]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> 2 = 1 + 1
[4,1,3,2] => [2,1,1]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> 1 = 0 + 1
[4,3,1,2] => [2,1,1]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> 1 = 0 + 1
[3,4,5,2,1] => [3,1,1]
=> 2 = 1 + 1
[3,5,2,4,1] => [2,2,1]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> 1 = 0 + 1
[3,5,4,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[4,2,5,3,1] => [2,2,1]
=> 1 = 0 + 1
[4,3,2,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[4,3,5,1,2] => [2,2,1]
=> 1 = 0 + 1
[4,3,5,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[4,5,1,3,2] => [2,2,1]
=> 1 = 0 + 1
[4,5,2,1,3] => [2,2,1]
=> 1 = 0 + 1
[4,5,2,3,1] => [2,2,1]
=> 1 = 0 + 1
[4,5,3,1,2] => [2,2,1]
=> 1 = 0 + 1
[4,5,3,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,1,4,3,2] => [2,1,1,1]
=> 1 = 0 + 1
[5,2,3,4,1] => [3,1,1]
=> 2 = 1 + 1
[5,2,4,1,3] => [2,2,1]
=> 1 = 0 + 1
[5,2,4,3,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,3,1,4,2] => [2,2,1]
=> 1 = 0 + 1
[5,3,2,1,4] => [2,1,1,1]
=> 1 = 0 + 1
[5,3,2,4,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,3,4,1,2] => [2,2,1]
=> 1 = 0 + 1
[5,3,4,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,4,1,2,3] => [3,1,1]
=> 2 = 1 + 1
[5,4,1,3,2] => [2,1,1,1]
=> 1 = 0 + 1
[5,4,2,1,3] => [2,1,1,1]
=> 1 = 0 + 1
Description
The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition.
Equivalently, given an integer partition $\lambda$, this is the number of molecular combinatorial species that decompose into a product of atomic species of sizes $\lambda_1,\lambda_2,\dots$. In particular, the value on the partition $(n)$ is the number of atomic species of degree $n$, [2].
Matching statistic: St001934
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001934: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 0 + 1
[1,2] => [2]
=> 1 = 0 + 1
[2,1] => [1,1]
=> 1 = 0 + 1
[1,2,3] => [3]
=> 2 = 1 + 1
[1,3,2] => [2,1]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> 1 = 0 + 1
[2,3,1] => [2,1]
=> 1 = 0 + 1
[3,1,2] => [2,1]
=> 1 = 0 + 1
[3,2,1] => [1,1,1]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> 1 = 0 + 1
[2,3,4,1] => [3,1]
=> 2 = 1 + 1
[2,4,1,3] => [2,2]
=> 1 = 0 + 1
[2,4,3,1] => [2,1,1]
=> 1 = 0 + 1
[3,1,4,2] => [2,2]
=> 1 = 0 + 1
[3,2,1,4] => [2,1,1]
=> 1 = 0 + 1
[3,2,4,1] => [2,1,1]
=> 1 = 0 + 1
[3,4,1,2] => [2,2]
=> 1 = 0 + 1
[3,4,2,1] => [2,1,1]
=> 1 = 0 + 1
[4,1,2,3] => [3,1]
=> 2 = 1 + 1
[4,1,3,2] => [2,1,1]
=> 1 = 0 + 1
[4,2,1,3] => [2,1,1]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> 1 = 0 + 1
[4,3,1,2] => [2,1,1]
=> 1 = 0 + 1
[4,3,2,1] => [1,1,1,1]
=> 1 = 0 + 1
[2,5,4,3,1] => [2,1,1,1]
=> 1 = 0 + 1
[3,4,5,2,1] => [3,1,1]
=> 2 = 1 + 1
[3,5,2,4,1] => [2,2,1]
=> 1 = 0 + 1
[3,5,4,1,2] => [2,2,1]
=> 1 = 0 + 1
[3,5,4,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[4,2,5,3,1] => [2,2,1]
=> 1 = 0 + 1
[4,3,2,5,1] => [2,1,1,1]
=> 1 = 0 + 1
[4,3,5,1,2] => [2,2,1]
=> 1 = 0 + 1
[4,3,5,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[4,5,1,3,2] => [2,2,1]
=> 1 = 0 + 1
[4,5,2,1,3] => [2,2,1]
=> 1 = 0 + 1
[4,5,2,3,1] => [2,2,1]
=> 1 = 0 + 1
[4,5,3,1,2] => [2,2,1]
=> 1 = 0 + 1
[4,5,3,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,1,4,3,2] => [2,1,1,1]
=> 1 = 0 + 1
[5,2,3,4,1] => [3,1,1]
=> 2 = 1 + 1
[5,2,4,1,3] => [2,2,1]
=> 1 = 0 + 1
[5,2,4,3,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,3,1,4,2] => [2,2,1]
=> 1 = 0 + 1
[5,3,2,1,4] => [2,1,1,1]
=> 1 = 0 + 1
[5,3,2,4,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,3,4,1,2] => [2,2,1]
=> 1 = 0 + 1
[5,3,4,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[5,4,1,2,3] => [3,1,1]
=> 2 = 1 + 1
[5,4,1,3,2] => [2,1,1,1]
=> 1 = 0 + 1
[5,4,2,1,3] => [2,1,1,1]
=> 1 = 0 + 1
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St000496
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ 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
[1] => [1] => {{1}}
=> 0
[1,2] => [2,1] => {{1,2}}
=> 0
[2,1] => [1,2] => {{1},{2}}
=> 0
[1,2,3] => [3,2,1] => {{1,3},{2}}
=> 1
[1,3,2] => [2,3,1] => {{1,2,3}}
=> 0
[2,1,3] => [3,1,2] => {{1,2,3}}
=> 0
[2,3,1] => [1,3,2] => {{1},{2,3}}
=> 0
[3,1,2] => [2,1,3] => {{1,2},{3}}
=> 0
[3,2,1] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,4,3,2] => [2,3,4,1] => {{1,2,3,4}}
=> 0
[2,3,4,1] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[2,4,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 0
[2,4,3,1] => [1,3,4,2] => {{1},{2,3,4}}
=> 0
[3,1,4,2] => [2,4,1,3] => {{1,2,3,4}}
=> 0
[3,2,1,4] => [4,1,2,3] => {{1,2,3,4}}
=> 0
[3,2,4,1] => [1,4,2,3] => {{1},{2,3,4}}
=> 0
[3,4,1,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 0
[3,4,2,1] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[4,1,2,3] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[4,1,3,2] => [2,3,1,4] => {{1,2,3},{4}}
=> 0
[4,2,1,3] => [3,1,2,4] => {{1,2,3},{4}}
=> 0
[4,2,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[4,3,1,2] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[4,3,2,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,5,4,3,1] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 0
[3,4,5,2,1] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[3,5,2,4,1] => [1,4,2,5,3] => {{1},{2,3,4,5}}
=> 0
[3,5,4,1,2] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 0
[3,5,4,2,1] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 0
[4,2,5,3,1] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 0
[4,3,2,5,1] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 0
[4,3,5,1,2] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 0
[4,3,5,2,1] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 0
[4,5,1,3,2] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 0
[4,5,2,1,3] => [3,1,2,5,4] => {{1,2,3},{4,5}}
=> 0
[4,5,2,3,1] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 0
[4,5,3,1,2] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 0
[4,5,3,2,1] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
[5,1,4,3,2] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 0
[5,2,3,4,1] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => {{1,2,3,4},{5}}
=> 0
[5,2,4,3,1] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 0
[5,3,1,4,2] => [2,4,1,3,5] => {{1,2,3,4},{5}}
=> 0
[5,3,2,1,4] => [4,1,2,3,5] => {{1,2,3,4},{5}}
=> 0
[5,3,2,4,1] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 0
[5,3,4,1,2] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 0
[5,3,4,2,1] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 0
[5,4,1,2,3] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 1
[5,4,1,3,2] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 0
[5,4,2,1,3] => [3,1,2,4,5] => {{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: St000660
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[4,5,1,3,2] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[5,1,4,3,2] => [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St001033
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 0
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 0
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 0
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 0
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 0
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[3,5,4,1,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 0
[3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[4,5,1,3,2] => [[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[5,1,4,3,2] => [[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0
[5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[5,3,1,4,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
=> [1,0,1,1,0,0,1,0,1,0]
=> 0
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
Matching statistic: St001137
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 0
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 0
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0
[1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 0
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0
[3,1,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0
[3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 0
[3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 0
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 0
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[2,5,4,3,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,4,5,2,1] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,5,2,4,1] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 0
[3,5,4,1,2] => [1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> 0
[3,5,4,2,1] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,2,5,3,1] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,3,2,5,1] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[4,3,5,1,2] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[4,3,5,2,1] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[4,5,1,3,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,5,2,1,3] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[4,5,2,3,1] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 0
[4,5,3,1,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[4,5,3,2,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[5,1,4,3,2] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[5,2,4,1,3] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 0
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[5,3,1,4,2] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[5,3,2,1,4] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[5,3,2,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 0
[5,3,4,1,2] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[5,3,4,2,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 0
[5,4,1,2,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[5,4,1,3,2] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[5,4,2,1,3] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St001311
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001311: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2,1] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([],2)
=> 0
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> 0
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> 0
[3,2,1] => [1,2,3] => ([],3)
=> 0
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> 0
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> 0
[4,3,2,1] => [1,2,3,4] => ([],4)
=> 0
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 0
[3,5,4,2,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[4,2,5,3,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[4,3,2,5,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,3,5,1,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[4,3,5,2,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[4,5,1,3,2] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[4,5,2,1,3] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[4,5,2,3,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0
[4,5,3,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[5,1,4,3,2] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 0
[5,2,4,3,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[5,3,1,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 0
[5,3,2,1,4] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5,3,2,4,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[5,3,4,1,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0
[5,3,4,2,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
[5,4,1,2,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[5,4,1,3,2] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0
[5,4,2,1,3] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 0
Description
The cyclomatic number of a graph.
This is the minimum number of edges that must be removed from the graph so that the result is a forest. This is also the first Betti number of the graph. It can be computed as $c + m - n$, where $c$ is the number of connected components, $m$ is the number of edges and $n$ is the number of vertices.
Matching statistic: St001317
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St001317: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [2,1] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([],2)
=> 0
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> 0
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> 0
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> 0
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> 0
[3,2,1] => [1,2,3] => ([],3)
=> 0
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 0
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 0
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 0
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 0
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 0
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 0
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> 0
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 0
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> 0
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> 0
[4,3,2,1] => [1,2,3,4] => ([],4)
=> 0
[2,5,4,3,1] => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,4,5,2,1] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 1
[3,5,2,4,1] => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> 0
[3,5,4,1,2] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 0
[3,5,4,2,1] => [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 0
[4,2,5,3,1] => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> 0
[4,3,2,5,1] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,3,5,1,2] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[4,3,5,2,1] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 0
[4,5,1,3,2] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[4,5,2,1,3] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 0
[4,5,2,3,1] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[4,5,3,1,2] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0
[4,5,3,2,1] => [1,2,3,5,4] => ([(3,4)],5)
=> 0
[5,1,4,3,2] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5,2,3,4,1] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[5,2,4,1,3] => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> 0
[5,2,4,3,1] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 0
[5,3,1,4,2] => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> 0
[5,3,2,1,4] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> 0
[5,3,2,4,1] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> 0
[5,3,4,1,2] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0
[5,3,4,2,1] => [1,2,4,3,5] => ([(3,4)],5)
=> 0
[5,4,1,2,3] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 1
[5,4,1,3,2] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 0
[5,4,2,1,3] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
=> 0
Description
The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph.
A graph is a forest if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,c)$ and $(b,c)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
The following 519 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. 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. St000920The logarithmic height of a Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St000002The number of occurrences of the pattern 123 in a permutation. St000052The number of valleys of a Dyck path not on the x-axis. 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. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000209Maximum difference of elements in cycles. 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. St000356The number of occurrences of the pattern 13-2. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000387The matching number of a graph. St000463The number of admissible inversions of a permutation. 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. St000535The rank-width of a graph. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000877The depth of the binary word interpreted as a path. St000884The number of isolated descents of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001071The beta invariant of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. 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. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001512The minimum rank of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001584The area statistic between a Dyck path and its bounce path. St001638The book thickness of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001928The number of non-overlapping descents in a permutation. St000040The number of regions of the inversion arrangement of a permutation. St000048The multinomial of the parts of a partition. St000058The order of a permutation. St000182The number of permutations whose cycle type is the given integer partition. St000201The number of leaf nodes in a binary tree. St000268The number of strongly connected orientations of a graph. St000292The number of ascents of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000396The register function (or Horton-Strahler number) of a binary tree. St000451The length of the longest pattern of the form k 1 2. St000453The number of distinct Laplacian eigenvalues of a graph. St000470The number of runs in a permutation. St000679The pruning number of an ordered tree. 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. St000862The number of parts of the shifted shape of a permutation. St000883The number of longest increasing subsequences of a permutation. 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. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. 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. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. 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. St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. 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. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000354The number of recoils of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. 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. St000624The normalized sum of the minimal distances to a greater element. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000795The mad of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St000485The length of the longest cycle of a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001246The maximal difference between two consecutive entries of a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St001718The number of non-empty open intervals in a poset. St000095The number of triangles of a graph. 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. St001596The number of two-by-two squares inside a skew partition. St001597The Frobenius rank of a skew partition. St000299The number of nonisomorphic vertex-induced subtrees. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001651The Frankl number of a lattice. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001592The maximal number of simple paths between any two different vertices of a graph. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001933The largest multiplicity of a part in an integer partition. 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 $3$. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000731The number of double exceedences of a permutation. St000648The number of 2-excedences of a permutation. St000647The number of big descents of a permutation. St001394The genus of a permutation. St000360The number of occurrences of the pattern 32-1. St001411The number of patterns 321 or 3412 in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St000358The number of occurrences of the pattern 31-2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000711The number of big exceedences of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St000223The number of nestings in the permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000538The number of even inversions of a permutation. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000836The number of descents of distance 2 of a permutation. St000886The number of permutations with the same antidiagonal sums. St000779The tier of a permutation. St000649The number of 3-excedences of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St000934The 2-degree of an integer partition. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001115The number of even descents of a permutation. St000367The number of simsun double descents of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000220The number of occurrences of the pattern 132 in a permutation. St000359The number of occurrences of the pattern 23-1. St000119The number of occurrences of the pattern 321 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000264The girth of a graph, which is not a tree. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000218The number of occurrences of the pattern 213 in a permutation. St000353The number of inner valleys of a permutation. St000534The number of 2-rises of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000078The number of alternating sign matrices whose left key is the permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000365The number of double ascents of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000145The Dyson rank of a partition. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer 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. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001498The normalised height of a Nakayama algebra with magnitude 1. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. 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. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. 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. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. 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. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements 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. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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. St001901The largest multiplicity of an irreducible representation 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. St001914The size of the orbit 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. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000806The semiperimeter of the associated bargraph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000357The number of occurrences of the pattern 12-3. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St000079The number of alternating sign matrices for a given Dyck path. 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. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000055The inversion sum of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000316The number of non-left-to-right-maxima of a permutation. St000355The number of occurrences of the pattern 21-3. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000663The number of right floats of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. 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. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001874Lusztig's a-function for the symmetric group. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000109The number of elements less than or equal to the given element in Bruhat order. St000325The width of the tree associated to a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000889The number of alternating sign matrices with the same antidiagonal sums. St000948The chromatic discriminant of a graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000216The absolute length of a permutation. St000646The number of big ascents of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001859The number of factors of the Stanley symmetric function associated with a permutation. 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. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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. St000219The number of occurrences of the pattern 231 in a permutation. St000379The number of Hamiltonian cycles in a graph. St001281The normalized isoperimetric number of a graph. 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. 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. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001875The number of simple modules with projective dimension at most 1. St000117The number of centered tunnels of a Dyck path. St000348The non-inversion sum of a binary word. St000628The balance of a binary word. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001910The height of the middle non-run of a Dyck path. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000390The number of runs of ones in a binary word. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000675The number of centered multitunnels of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000983The length of the longest alternating subword. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001595The number of standard Young tableaux of the skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St000015The number of peaks of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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 $eA$. St001202Call 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. St001203We associate to 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 Dyck path as follows:
St001530The depth of a Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. 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 $eA$. St001060The distinguishing index of a graph. St001118The acyclic chromatic index of a graph. St000456The monochromatic index of a connected graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001684The reduced word complexity of a permutation. St001871The number of triconnected components of a graph. St001811The Castelnuovo-Mumford regularity of a permutation. St001569The maximal modular displacement of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001570The minimal number of edges to add to make a graph Hamiltonian. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001866The nesting alignments 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000741The Colin de Verdière graph invariant. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001630The global dimension of the incidence algebra of the lattice over the rational numbers.
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