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Your data matches 103 different statistics following compositions of up to 3 maps.
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Matching statistic: St000051
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> 0
[.,[.,.]]
=> 0
[[.,.],.]
=> 1
[.,[.,[.,.]]]
=> 0
[.,[[.,.],.]]
=> 0
[[.,.],[.,.]]
=> 1
[[.,[.,.]],.]
=> 2
[[[.,.],.],.]
=> 2
[.,[.,[.,[.,.]]]]
=> 0
[.,[.,[[.,.],.]]]
=> 0
[.,[[.,.],[.,.]]]
=> 0
[.,[[.,[.,.]],.]]
=> 0
[.,[[[.,.],.],.]]
=> 0
[[.,.],[.,[.,.]]]
=> 1
[[.,.],[[.,.],.]]
=> 1
[[.,[.,.]],[.,.]]
=> 2
[[[.,.],.],[.,.]]
=> 2
[[.,[.,[.,.]]],.]
=> 3
[[.,[[.,.],.]],.]
=> 3
[[[.,.],[.,.]],.]
=> 3
[[[.,[.,.]],.],.]
=> 3
[[[[.,.],.],.],.]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> 0
[.,[.,[[.,.],[.,.]]]]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> 0
[.,[.,[[[.,.],.],.]]]
=> 0
[.,[[.,.],[.,[.,.]]]]
=> 0
[.,[[.,.],[[.,.],.]]]
=> 0
[.,[[.,[.,.]],[.,.]]]
=> 0
[.,[[[.,.],.],[.,.]]]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> 0
[.,[[.,[[.,.],.]],.]]
=> 0
[.,[[[.,.],[.,.]],.]]
=> 0
[.,[[[.,[.,.]],.],.]]
=> 0
[.,[[[[.,.],.],.],.]]
=> 0
[[.,.],[.,[.,[.,.]]]]
=> 1
[[.,.],[.,[[.,.],.]]]
=> 1
[[.,.],[[.,.],[.,.]]]
=> 1
[[.,.],[[.,[.,.]],.]]
=> 1
[[.,.],[[[.,.],.],.]]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> 2
[[.,[.,.]],[[.,.],.]]
=> 2
[[[.,.],.],[.,[.,.]]]
=> 2
[[[.,.],.],[[.,.],.]]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> 3
[[.,[[.,.],.]],[.,.]]
=> 3
[[[.,.],[.,.]],[.,.]]
=> 3
[[[.,[.,.]],.],[.,.]]
=> 3
[[[[.,.],.],.],[.,.]]
=> 3
Description
The size of the left subtree of a binary tree.
Matching statistic: St000026
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [1,1,0,0]
=> 2 = 1 + 1
[[.,.],.]
=> [1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5 = 4 + 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5 = 4 + 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 4 + 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000501
(load all 109 compositions to match this statistic)
(load all 109 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000501: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000501: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => 3 = 2 + 1
[[.,.],[.,.]]
=> [1,3,2] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 4 = 3 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 5 = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 5 = 4 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 5 = 4 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 5 = 4 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 5 = 4 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1 = 0 + 1
Description
The size of the first part in the decomposition of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is defined to be the smallest $k > 0$ such that $\{\pi(1),\ldots,\pi(k)\} = \{1,\ldots,k\}$. This statistic is undefined for the empty permutation.
For the number of parts in the decomposition see [[St000056]].
Matching statistic: St000740
(load all 114 compositions to match this statistic)
(load all 114 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => 2 = 1 + 1
[.,[.,[.,.]]]
=> [3,2,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [3,1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => 3 = 2 + 1
[[[.,.],.],.]
=> [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 4 = 3 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 4 = 3 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 4 = 3 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 4 = 3 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 4 = 3 + 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000133
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000133: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => 0
[[.,.],.]
=> [1,2] => [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 2
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,3,2] => 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,4,3,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,4,3,1] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => 3
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 3
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,4,3,2] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,4,1,3] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,4,3,2] => 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,4,3] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,4,3,2] => 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,5,4,2,1] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,5,4,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,5,4,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,5,4,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,5,4,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,5,4,3,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,5,2,4,1] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,5,4,3,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,5,4,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,5,4,3,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => 4
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,3,2] => 4
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,4,3,2] => 4
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => 4
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,5,4,3,2] => 4
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,5,4,3,2] => 4
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,5,4,3,2] => 4
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,5,4,3] => 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,5,4,3,2] => 4
Description
The "bounce" of a permutation.
Matching statistic: St000645
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000645: 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
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 3
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St001225
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001225: 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
St001225: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 0
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 0
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 2
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 4
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 4
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 4
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000054
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00325: Permutations —ones to leading⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 3 = 2 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,1,2] => 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,4,1] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,3,2,1] => 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,4,3] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,2,1,3] => 4 = 3 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,3,1] => 4 = 3 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,3,2] => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,4,2] => 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,2,4] => 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,1,2,3] => 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,5,1,2] => 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,1,3] => 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,2,4,5,1] => 3 = 2 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,5,4,1] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,1,3,2] => 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,3,5,2,1] => 4 = 3 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,4,3,1,2] => 5 = 4 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,4,3,2,1] => 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,1,3,4,5] => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,1,4,3,5] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,1,3,5,4] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,5,3,4] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,5,4,3] => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,3,1,2,4] => 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,2,5,1,3] => 4 = 3 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,2,1,4] => 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,4,5,2] => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,1,5,4,2] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,3,1,4,2] => 5 = 4 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,5,3,1] => 4 = 3 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,2,1,4,3] => 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,5,3,2] => 4 = 3 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,4,2,1] => 5 = 4 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,3,4,1,2] => 5 = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,2,4,3,1] => 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,4,1,3] => 5 = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,2,3] => 5 = 4 + 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000066
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [[1]]
=> 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [[0,1],[1,0]]
=> 2 = 1 + 1
[[.,.],.]
=> [1,2] => [[1,0],[0,1]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3 = 2 + 1
[[.,.],[.,.]]
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 4 = 3 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4 = 3 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1 = 0 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
=> 5 = 4 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
=> 5 = 4 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 5 = 4 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 5 = 4 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 5 = 4 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]]
=> 5 = 4 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 5 = 4 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 5 = 4 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 5 = 4 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3 = 2 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1 = 0 + 1
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
Matching statistic: St000335
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000335: 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
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1,0]
=> 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1 = 0 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
Description
The difference of lower and upper interactions.
An ''upper interaction'' in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
The following 93 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000382The first part of an integer composition. St000383The last part of an integer composition. St000505The biggest entry in the block containing the 1. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001481The minimal height of a peak of a Dyck path. 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. St000019The cardinality of the support of a permutation. St000030The sum of the descent differences of a permutations. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000316The number of non-left-to-right-maxima of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000546The number of global descents of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000056The decomposition (or block) number of a permutation. St000240The number of indices that are not small excedances. St000273The domination number of a graph. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000700The protection number of an ordered tree. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000883The number of longest increasing subsequences of a permutation. St000916The packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001461The number of topologically connected components of the chord diagram of a permutation. St001497The position of the largest weak excedence of a permutation. St001725The harmonious chromatic number of a graph. St001829The common independence number of a graph. St000439The position of the first down step of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000297The number of leading ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St000795The mad of a permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000989The number of final rises of a permutation. St001480The number of simple summands of the module J^2/J^3. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000784The maximum of the length and the largest part of the integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000225Difference between largest and smallest parts in a partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000840The number of closers smaller than the largest opener in a perfect matching. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001434The number of negative sum pairs of a signed permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000454The largest eigenvalue of a graph if it is integral. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001645The pebbling number of a connected graph. St001557The number of inversions of the second entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000455The second largest eigenvalue of a graph if it is integral.
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