Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000734
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1] => [[1]]
 => 1
[[.,.],.]
 => [1,2] => [1] => [[1]]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,1] => [[1],[2]]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [2,1] => [[1],[2]]
 => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,2] => [[1,2]]
 => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1] => [[1],[2]]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2] => [[1,2]]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,2,1] => [[1],[2],[3]]
 => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,3,1] => [[1,2],[3]]
 => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,1] => [[1],[2],[3]]
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,1] => [[1,2],[3]]
 => 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,2] => [[1,2],[3]]
 => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,3,2] => [[1,2],[3]]
 => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,3] => [[1,3],[2]]
 => 3
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3] => [[1,2,3]]
 => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1] => [[1],[2],[3]]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1] => [[1,2],[3]]
 => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2] => [[1,2],[3]]
 => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3] => [[1,3],[2]]
 => 3
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3] => [[1,2,3]]
 => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 3
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [2,3,4,1] => [[1,2,3],[4]]
 => 3
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,1] => [[1,2],[3],[4]]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [2,4,3,1] => [[1,2],[3],[4]]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,1] => [[1,3],[2],[4]]
 => 3
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,1] => [[1,2,3],[4]]
 => 3
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,3,4,2] => [[1,2,3],[4]]
 => 3
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,3,4,2] => [[1,2,3],[4]]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,4,3] => [[1,3],[2,4]]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,4,3] => [[1,2,3],[4]]
 => 3
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,4] => [[1,4],[2],[3]]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,1,4] => [[1,2,4],[3]]
 => 4
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,4] => [[1,3,4],[2]]
 => 4
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4] => [[1,2,3,4]]
 => 4
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000740
(load all 3 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1] => [1] => 1
[[.,.],.]
 => [1,2] => [1] => [1] => 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,1] => [2,1] => 1
[.,[[.,.],.]]
 => [2,3,1] => [2,1] => [2,1] => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,2] => [1,2] => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1] => [2,1] => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2] => [1,2] => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1] => [3,2,1] => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,2,1] => [3,2,1] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,3,1] => [3,1,2] => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,1] => [3,2,1] => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,2] => [1,3,2] => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,3,2] => [1,3,2] => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,3] => [2,1,3] => 3
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1] => [3,2,1] => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2] => [1,3,2] => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3] => [2,1,3] => 3
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,2,1] => [4,3,1,2] => 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2,4,1] => [4,2,1,3] => 3
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,1] => [4,3,1,2] => 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,1] => [4,2,1,3] => 3
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 3
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 3
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 4
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 1
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St001291
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1] => [1,0]
 => 1
[[.,.],.]
 => [1,2] => [1] => [1,0]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,1] => [1,1,0,0]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [2,1] => [1,1,0,0]
 => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,2] => [1,0,1,0]
 => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1] => [1,1,0,0]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2] => [1,0,1,0]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
 => 3
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
 => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
 => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
 => 3
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
 => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 3
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 3
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 3
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 4
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 4
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 4
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 1
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000019
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1,2] => [1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [2,1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [2,1,3] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1,2,4] => [3,1,2] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [3,2,1,4] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,1,3] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
Description
The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.
Matching statistic: St000141
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1,2] => [1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [2,1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [1,3,2] => [2,3,1] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,3,4,2] => [1,3,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,3,4,1] => [2,3,1] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [2,4,3,1] => [2,3,1] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [3,4,1,2] => [3,1,2] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [3,4,2,1] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,3] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,2] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [1,3,5,4,2] => [1,3,4,2] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [1,4,5,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [1,4,5,3,2] => [1,4,3,2] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [3,4,5,1,2] => [3,4,1,2] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,5,4,1,2] => [3,4,1,2] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [4,5,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [4,5,1,3,2] => [4,1,3,2] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [4,5,3,1,2] => [4,3,1,2] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000316
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1,2] => [1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [2,1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [2,1,3] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1,2,4] => [3,1,2] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [3,2,1,4] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,1,3] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
Description
The number of non-left-to-right-maxima of a permutation. An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St001300
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1] => ([],1)
 => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [1] => ([],1)
 => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,1] => ([],2)
 => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [2,1] => ([],2)
 => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [1,2] => ([(0,1)],2)
 => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1] => ([],2)
 => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2] => ([(0,1)],2)
 => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1] => ([],3)
 => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,2,1] => ([],3)
 => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,3,1] => ([(1,2)],3)
 => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,1] => ([],3)
 => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,1] => ([(1,2)],3)
 => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1] => ([],3)
 => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1] => ([(1,2)],3)
 => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,2] => ([(1,2)],3)
 => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
 => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
 => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
 => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1] => ([],4)
 => 0 = 1 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000727
(load all 2 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1] => [1] => ? = 1
[[.,.],.]
 => [1,2] => [1] => [1] => ? = 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,1] => [2,1] => 1
[.,[[.,.],.]]
 => [2,3,1] => [2,1] => [2,1] => 1
[[.,.],[.,.]]
 => [1,3,2] => [1,2] => [1,2] => 2
[[.,[.,.]],.]
 => [2,1,3] => [2,1] => [2,1] => 1
[[[.,.],.],.]
 => [1,2,3] => [1,2] => [1,2] => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1] => [3,2,1] => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,2,1] => [3,2,1] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,3,1] => [3,1,2] => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [3,2,1] => [3,2,1] => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,2] => [1,3,2] => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,3,2] => [1,3,2] => 2
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,3] => [2,1,3] => 3
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 3
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,2,1] => [3,2,1] => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2] => [1,3,2] => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3] => [2,1,3] => 3
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 3
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [3,4,2,1] => [4,3,1,2] => 2
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,4,2,1] => [4,3,1,2] => 2
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [3,2,4,1] => [4,2,1,3] => 3
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [3,4,2,1] => [4,3,1,2] => 2
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [2,4,3,1] => [4,1,3,2] => 2
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [3,2,4,1] => [4,2,1,3] => 3
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => 2
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,3,4,2] => [1,4,2,3] => 3
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => 2
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 3
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 3
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 3
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 4
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 4
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 4
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [3,4,2,1] => [4,3,1,2] => 2
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [2,4,3,1] => [4,1,3,2] => 2
Description
The largest label of a leaf in the binary search tree associated with the permutation. Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.
Matching statistic: St000653
(load all 3 compositions to match this statistic)
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [1,1,0,0]
 => [1,2] => [1] => ? = 1 - 1
[[.,.],.]
 => [1,0,1,0]
 => [2,1] => [1] => ? = 1 - 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,2,3] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [3,1,2] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [2,1,3] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,3,2] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [2,3,1] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,3,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,1,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,3,2] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,3,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => [4,1,2,3] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,1,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [3,1,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,2,3,5] => [1,4,2,3] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,4,1,2] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,1,5,2,3] => [4,1,2,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [3,1,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [3,4,1,2] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [2,4,1,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [2,3,1,4] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [1,2,4,3] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [3,1,4,2] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,3,4,2] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [2,3,4,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => [4,1,2,3] => 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,1,2,4] => 1 = 2 - 1
Description
The last descent of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000957
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
 => [2,1] => [1,2] => [1] => ? = 1 - 1
[[.,.],.]
 => [1,2] => [2,1] => [1] => ? = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,2,3] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [1,3,2] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [2,1,3] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [3,1,2] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [3,2,1] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,3,2,4] => [1,3,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,1,2,4] => [3,1,2] => 2 = 3 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [3,2,1,4] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [4,2,1,3] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2 = 3 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [5,1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [5,1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
Description
The number of Bruhat lower covers of a permutation. This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$. This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000147The largest part of an integer partition. St001280The number of parts of an integer partition that are at least two. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000327The number of cover relations in a poset. St001557The number of inversions of the second entry of a permutation.