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Your data matches 99 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 = 1 - 1
[.,[.,.]]
=> 0 = 1 - 1
[[.,.],.]
=> 1 = 2 - 1
[.,[.,[.,.]]]
=> 0 = 1 - 1
[.,[[.,.],.]]
=> 0 = 1 - 1
[[.,.],[.,.]]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> 2 = 3 - 1
[[[.,.],.],.]
=> 2 = 3 - 1
[.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> 0 = 1 - 1
[[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[[.,[[.,.],.]],.]
=> 3 = 4 - 1
[[[.,.],[.,.]],.]
=> 3 = 4 - 1
[[[.,[.,.]],.],.]
=> 3 = 4 - 1
[[[[.,.],.],.],.]
=> 3 = 4 - 1
[.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
=> 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
=> 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
=> 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
=> 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> 3 = 4 - 1
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
[.,[.,.]]
=> [1,1,0,0]
=> 2
[[.,.],.]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 3
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 4
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 4
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 4
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 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
[.,[.,.]]
=> [2,1] => 2
[[.,.],.]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => 3
[.,[[.,.],.]]
=> [2,3,1] => 3
[[.,.],[.,.]]
=> [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 2
[[[.,.],.],.]
=> [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 4
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 4
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 3
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 5
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 5
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 5
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 5
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 5
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 5
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 5
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 5
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 5
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 5
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 5
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 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
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 2
[.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => 2
[[.,[.,.]],.]
=> [2,1,3] => 3
[[[.,.],.],.]
=> [1,2,3] => 3
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 3
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 3
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 4
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 4
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 4
[[[[.,.],.],.],.]
=> [1,2,3,4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 3
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 3
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 3
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 3
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 4
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 4
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 4
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 4
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 4
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
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
[.,[.,.]]
=> [2,1] => [2,1] => 2
[[.,.],.]
=> [1,2] => [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 3
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 2
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 3
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,1,2] => 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,4,1] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,3,2,1] => 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,4,3] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,2,1,3] => 4
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 3
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,3,1] => 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => 4
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,3,2] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,4,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,2,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,1,2,3] => 5
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,5,1,2] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,1,3] => 5
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,2,4,5,1] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,5,4,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,1,3,2] => 5
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,3,5,2,1] => 4
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,4,3,1,2] => 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,4,3,2,1] => 5
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,1,3,4,5] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,1,4,3,5] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,1,3,5,4] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,5,3,4] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,5,4,3] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,3,1,2,4] => 5
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,2,5,1,3] => 4
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,2,1,4] => 5
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,4,5,2] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,1,5,4,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,3,1,4,2] => 5
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,5,3,1] => 4
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,2,1,4,3] => 5
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,5,3,2] => 4
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,4,2,1] => 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,3,4,1,2] => 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,2,4,3,1] => 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,4,1,3] => 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,2,3] => 5
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
[.,[.,.]]
=> [2,1] => [[0,1],[1,0]]
=> 2
[[.,.],.]
=> [1,2] => [[1,0],[0,1]]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3
[[.,.],[.,.]]
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 4
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 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
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [[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
[.,[.,[[.,.],[.,.]]]]
=> [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
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [[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
[.,[.,[[[.,.],.],.]]]
=> [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
[.,[[.,.],[.,[.,.]]]]
=> [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
[.,[[.,.],[[.,.],.]]]
=> [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
[.,[[.,[.,.]],[.,.]]]
=> [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
[.,[[[.,.],.],[.,.]]]
=> [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
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [[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
[.,[[.,[[.,.],.]],.]]
=> [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
[.,[[[.,.],[.,.]],.]]
=> [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
[.,[[[.,[.,.]],.],.]]
=> [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
[.,[[[[.,.],.],.],.]]
=> [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
[[.,.],[.,[.,[.,.]]]]
=> [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
[[.,.],[.,[[.,.],.]]]
=> [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
[[.,.],[[.,.],[.,.]]]
=> [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
[[.,.],[[.,[.,.]],.]]
=> [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
[[.,.],[[[.,.],.],.]]
=> [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
[[.,[.,.]],[.,[.,.]]]
=> [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
[[.,[.,.]],[[.,.],.]]
=> [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
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [[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
[[[.,.],.],[[.,.],.]]
=> [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
[[.,[.,[.,.]]],[.,.]]
=> [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
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [[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
[[[.,.],[.,.]],[.,.]]
=> [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
[[[.,[.,.]],.],[.,.]]
=> [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
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [[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
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
[.,[.,.]]
=> [2,1] => [1,1,0,0]
=> 2
[[.,.],.]
=> [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[[.,.],.]]
=> [2,3,1] => [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[[.,[.,.]],.]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],.]]]
=> [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]
=> 4
[.,[[.,[.,.]],.]]
=> [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]
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[[.,.],[[.,.],.]]
=> [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]
=> 4
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[[.,[.,[.,.]]],.]
=> [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]
=> 1
[[[.,.],[.,.]],.]
=> [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]
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [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
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [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]
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [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]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,.],[[.,.],.]]]
=> [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]
=> 5
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [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]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [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]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,.],[.,[[.,.],.]]]
=> [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]
=> 5
[[.,.],[[.,[.,.]],.]]
=> [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]
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[.,[.,.]],[[.,.],.]]
=> [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]
=> 5
[[[.,.],.],[[.,.],.]]
=> [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]
=> 5
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
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.
Matching statistic: St000382
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 1
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 1
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 3
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 3
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 3
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 3
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 4
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 4
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 4
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 4
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 4
Description
The first part of an integer composition.
Matching statistic: St000383
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 1
[.,[.,.]]
=> [1,0,1,0]
=> [1,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => 2
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => 3
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 3
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 3
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 3
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 4
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 4
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 4
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 4
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000505
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> {{1}}
=> 1
[.,[.,.]]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[[.,.],.]
=> [1,0,1,0]
=> {{1},{2}}
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 3
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 4
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 4
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 4
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 5
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 5
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 5
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 5
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 5
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 5
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 5
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 5
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 5
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 5
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 5
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 5
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 5
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
Description
The biggest entry in the block containing the 1.
The following 89 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000133The "bounce" of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. 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. 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. St000439The position of the first down step of a Dyck path. 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. St000326The position of the first one in a binary word after appending a 1 at the end. 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. 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. 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. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000840The number of closers smaller than the largest opener in a perfect matching. 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. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000456The monochromatic index of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001557The number of inversions of the second entry of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice.
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