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Your data matches 633 different statistics following compositions of up to 3 maps.
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Matching statistic: St000121
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
St000121: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> 0
[[.,.],.]
=> 0
[.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> 0
[[.,.],[.,.]]
=> 0
[[.,[.,.]],.]
=> 0
[[[.,.],.],.]
=> 0
[.,[.,[.,[.,.]]]]
=> 1
[.,[.,[[.,.],.]]]
=> 0
[.,[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],.]]
=> 0
[.,[[[.,.],.],.]]
=> 0
[[.,.],[.,[.,.]]]
=> 0
[[.,.],[[.,.],.]]
=> 0
[[.,[.,.]],[.,.]]
=> 0
[[[.,.],.],[.,.]]
=> 0
[[.,[.,[.,.]]],.]
=> 0
[[.,[[.,.],.]],.]
=> 0
[[[.,.],[.,.]],.]
=> 0
[[[.,[.,.]],.],.]
=> 0
[[[[.,.],.],.],.]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> 1
[.,[[.,.],[[.,.],.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> 0
[.,[[[[.,.],.],.],.]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> 0
[[.,.],[[.,.],[.,.]]]
=> 0
[[.,.],[[.,[.,.]],.]]
=> 0
[[.,.],[[[.,.],.],.]]
=> 0
[[.,[.,.]],[.,[.,.]]]
=> 0
[[.,[.,.]],[[.,.],.]]
=> 0
[[[.,.],.],[.,[.,.]]]
=> 0
[[[.,.],.],[[.,.],.]]
=> 0
[[.,[.,[.,.]]],[.,.]]
=> 0
[[.,[[.,.],.]],[.,.]]
=> 0
[[[.,.],[.,.]],[.,.]]
=> 0
[[[.,[.,.]],.],[.,.]]
=> 0
[[[[.,.],.],.],[.,.]]
=> 0
[[.,[.,[.,[.,.]]]],.]
=> 1
Description
The number of occurrences of the contiguous pattern {{{[.,[.,[.,[.,.]]]]}}} in a binary tree.
[[oeis:A036765]] counts binary trees avoiding this pattern.
Matching statistic: St001520
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001520: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1
Description
The number of strict 3-descents.
A '''strict 3-descent''' of a permutation $\pi$ of $\{1,2, \dots ,n \}$ is a pair $(i,i+3)$ with $ i+3 \leq n$ and $\pi(i) > \pi(i+3)$.
Matching statistic: St000034
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St000119
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000119: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of occurrences of the pattern 321 in a permutation.
Matching statistic: St000367
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000367: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000367: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of simsun double descents of a permutation.
The restriction of a permutation $\pi$ to $[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of $\pi$ of letters in $[k]$.
A simsun double descent of a permutation $\pi$ is a double descent of any restriction of $\pi$ to $[1,\ldots,k]$ for some $k$. (Note here that the same double descent can appear in multiple restrictions!)
Matching statistic: St000404
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation.
A permutation avoids these two pattern if and only if it is an ''input-restricted deques'', see [1].
Matching statistic: St000406
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000406: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000406: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of occurrences of the pattern 3241 in a permutation.
Matching statistic: St000437
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000437: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000437: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of occurrences of the pattern 312 or of the pattern 321 in a permutation.
Matching statistic: St000648
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of 2-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+2$.
Matching statistic: St000711
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
Description
The number of big exceedences of a permutation.
A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$.
This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
The following 623 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St001411The number of patterns 321 or 3412 in a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St001490The number of connected components of a skew partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001568The smallest positive integer that does not appear twice in the partition. St000264The girth of a graph, which is not a tree. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St000260The radius of a connected graph. 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 $K[x]/(x^n)$. St001549The number of restricted non-inversions between exceedances. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000445The number of rises of length 1 of a Dyck path. 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 $eA$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000068The number of minimal elements in a poset. St001626The number of maximal proper sublattices of a lattice. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000268The number of strongly connected orientations of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000637The length of the longest cycle in a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001354The number of series nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001970The signature of a graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000456The monochromatic index of a connected graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000655The length of the minimal rise of a Dyck path. St000671The maximin edge-connectivity for choosing a subgraph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000948The chromatic discriminant of a graph. St001271The competition number of a graph. St001277The degeneracy of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001395The number of strictly unfriendly partitions of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001826The maximal number of leaves on a vertex of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000396The register function (or Horton-Strahler number) of a binary tree. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St000172The Grundy number of a graph. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001963The tree-depth of a graph. St000523The number of 2-protected nodes of a rooted tree. St000397The Strahler number of a rooted tree. St000700The protection number of an ordered tree. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. 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. St000678The number of up steps after the last double rise of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000444The length of the maximal rise of a Dyck path. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000439The position of the first down step of a Dyck path. St001875The number of simple modules with projective dimension at most 1. St000981The length of the longest zigzag subpath. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St001964The interval resolution global dimension of a poset. St000011The number of touch points (or returns) of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000069The number of maximal elements of a poset. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St000920The logarithmic height of a Dyck path. St001330The hat guessing number of a graph. St000022The number of fixed points of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001498The normalised height of a Nakayama algebra with magnitude 1. St001625The Möbius invariant of a lattice. St000617The number of global maxima of a Dyck path. St000701The protection number of a binary tree. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. 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 $eA$. St000124The cardinality of the preimage of the Simion-Schmidt map. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 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. St000215The number of adjacencies of a permutation, zero appended. St000909The number of maximal chains of maximal size in a poset. St000679The pruning number of an ordered tree. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000351The determinant of the adjacency matrix of a graph. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000536The pathwidth of a graph. St000640The rank of the largest boolean interval in a poset. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. St001333The cardinality of a minimal edge-isolating set of a graph. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001638The book thickness of a graph. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000061The number of nodes on the left branch of a binary tree. St000469The distinguishing number of a graph. St000537The cutwidth of a graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St000778The metric dimension of a graph. St001119The length of a shortest maximal path in a graph. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001270The bandwidth of a graph. St001399The distinguishing number of a poset. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001644The dimension of a graph. St001724The 2-packing differential of a graph. St001962The proper pathwidth of a graph. St000171The degree of the graph. St001112The 3-weak dynamic number of a graph. St001118The acyclic chromatic index of a graph. St001110The 3-dynamic chromatic number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001746The coalition number of a graph. St000039The number of crossings of a permutation. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000090The variation of a composition. St000095The number of triangles of a graph. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000214The number of adjacencies of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000312The number of leaves in a graph. St000315The number of isolated vertices of a graph. St000317The cycle descent number of a permutation. St000338The number of pixed points of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000407The number of occurrences of the pattern 2143 in a permutation. St000449The number of pairs of vertices of a graph with distance 4. St000462The major index minus the number of excedences of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000496The rcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000502The number of successions of a set partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000516The number of stretching pairs of a permutation. St000546The number of global descents of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000699The toughness times the least common multiple of 1,. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000850The number of 1/2-balanced pairs in a poset. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000943The number of spots the most unlucky car had to go further in a parking function. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000962The 3-shifted major index of a permutation. St000974The length of the trunk of an ordered tree. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000989The number of final rises of a permutation. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001057The Grundy value of the game of creating an independent set in a graph. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001114The number of odd descents of a permutation. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001306The number of induced paths on four vertices in a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001356The number of vertices in prime modules of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001394The genus of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. 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. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001691The number of kings in a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001781The interlacing number of a set partition. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001871The number of triconnected components of a graph. St001895The oddness of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St001972The a-number of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000037The sign of a permutation. St000056The decomposition (or block) number of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000096The number of spanning trees of a graph. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000253The crossing number of a set partition. St000255The number of reduced Kogan faces with the permutation as type. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000307The number of rowmotion orbits of a poset. St000310The minimal degree of a vertex of a graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000353The number of inner valleys of a permutation. St000383The last part of an integer composition. St000450The number of edges minus the number of vertices plus 2 of a graph. St000461The rix statistic of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000633The size of the automorphism group of a poset. St000654The first descent of a permutation. St000667The greatest common divisor of the parts of the partition. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000823The number of unsplittable factors of the set partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000872The number of very big descents of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000883The number of longest increasing subsequences of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000907The number of maximal antichains of minimal length in a poset. St000910The number of maximal chains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000990The first ascent of a permutation. St000993The multiplicity of the largest part of an integer partition. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump 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 $K[x]/(x^n)$. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. 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. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001344The neighbouring number of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001393The induced matching number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001463The number of distinct columns in the nullspace of a graph. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001828The Euler characteristic of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001949The rigidity index of a graph. St000024The number of double up and double down steps of a Dyck path. St000058The order of a permutation. St000092The number of outer peaks of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000099The number of valleys of a permutation, including the boundary. St000105The number of blocks in the set partition. St000244The cardinality of the automorphism group of a graph. St000251The number of nonsingleton blocks of a set partition. St000258The burning number of a graph. St000273The domination number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000364The exponent of the automorphism group of a graph. St000470The number of runs in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000822The Hadwiger number of the graph. St000836The number of descents of distance 2 of a permutation. St000843The decomposition number of a perfect matching. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000916The packing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001060The distinguishing index of a graph. St001062The maximal size of a block of a set partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001621The number of atoms of a lattice. St001654The monophonic hull number of a graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001672The restrained domination number of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001741The largest integer such that all patterns of this size are contained in the permutation. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. 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). St000443The number of long tunnels of a Dyck path. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. 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. St001512The minimum rank of a graph. St001670The connected partition number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type.
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