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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000734
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%
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)
(load all 3 compositions to match this statistic)
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%
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: St000727
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
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%
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: St000957
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1 = 2 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [2,1,3,4] => 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1] => [1,3,2,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [6,7,8,5,4,1,2,3] => [6,7,5,4,1,2,3] => [2,1,3,4,7,6,5] => ? = 4 - 1
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,1,2,3] => [5,6,7,4,1,2,3] => [3,2,1,4,7,6,5] => ? = 5 - 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,1,2,3] => ? => ? => ? = 5 - 1
[[[.,.],.],[[[.,[.,[.,.]]],.],.]]
=> [6,5,4,7,8,1,2,3] => [6,5,4,7,1,2,3] => [2,3,4,1,7,6,5] => ? = 6 - 1
[[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => ? => ? => ? = 6 - 1
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [5,4,6,7,8,1,2,3] => ? => ? => ? = 6 - 1
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,1,2,3] => [4,5,6,7,1,2,3] => [4,3,2,1,7,6,5] => ? = 6 - 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [8,6,7,5,1,2,3,4] => [6,7,5,1,2,3,4] => [2,1,3,7,6,5,4] => ? = 5 - 1
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> [6,7,8,5,1,2,3,4] => [6,7,5,1,2,3,4] => [2,1,3,7,6,5,4] => ? = 5 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [8,6,5,7,1,2,3,4] => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 6 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 6 - 1
[[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [6,7,5,8,1,2,3,4] => [6,7,5,1,2,3,4] => [2,1,3,7,6,5,4] => ? = 5 - 1
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,1,2,3,4] => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 6 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 6 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,1,2,5] => ? => ? => ? = 5 - 1
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [6,7,8,4,3,1,2,5] => [6,7,4,3,1,2,5] => [2,1,4,5,7,6,3] => ? = 6 - 1
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
=> [7,6,8,3,4,1,2,5] => ? => ? => ? = 5 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [2,1,5,4,7,6,3] => ? = 6 - 1
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,1,2,3,5] => ? => ? => ? = 5 - 1
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [6,7,8,4,1,2,3,5] => [6,7,4,1,2,3,5] => [2,1,4,7,6,5,3] => ? = 6 - 1
[[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => ? => ? => ? = 5 - 1
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [6,7,8,3,1,2,4,5] => [6,7,3,1,2,4,5] => [2,1,5,7,6,4,3] => ? = 6 - 1
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => ? = 6 - 1
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 6 - 1
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 6 - 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 - 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [6,7,3,4,1,2,5,8] => ? => ? => ? = 6 - 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [7,3,1,2,4,5,6,8] => [7,3,1,2,4,5,6] => [1,5,7,6,4,3,2] => ? = 6 - 1
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 6 - 1
[[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [6,4,5,1,2,3,7,8] => ? => ? => ? = 7 - 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [6,4,1,2,3,5,7,8] => [6,4,1,2,3,5,7] => [2,4,7,6,5,3,1] => ? = 7 - 1
[[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [5,3,1,2,4,6,7,8] => [5,3,1,2,4,6,7] => [3,5,7,6,4,2,1] => ? = 7 - 1
[[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [5,1,2,3,4,6,7,8] => [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => ? = 7 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => ? = 7 - 1
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 - 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.
Matching statistic: St000019
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000019: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●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
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [3,2,1,4,5,7,6] => ? = 4 - 1
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [3,2,1,4,7,6,5] => ? = 5 - 1
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? => ? = 4 - 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,1,2,3] => [3,2,1,5,4,8,7,6] => ? => ? = 5 - 1
[[[.,.],.],[[[.,[.,[.,.]]],.],.]]
=> [6,5,4,7,8,1,2,3] => [3,2,1,8,7,4,5,6] => [3,2,1,7,4,5,6] => ? = 6 - 1
[[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => [3,2,1,8,7,4,6,5] => ? => ? = 6 - 1
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [5,4,6,7,8,1,2,3] => [3,2,1,8,7,6,4,5] => ? => ? = 6 - 1
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,1,2,3] => [3,2,1,8,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 6 - 1
[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [8,7,6,5,3,1,2,4] => [4,2,1,3,5,6,7,8] => ? => ? = 4 - 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [4,3,2,1,5,7,6] => ? = 5 - 1
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> [6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [4,3,2,1,5,7,6] => ? = 5 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [4,3,2,1,6,5,7] => ? = 5 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,7] => ? = 5 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [8,6,5,7,1,2,3,4] => [4,3,2,1,7,5,6,8] => [4,3,2,1,7,5,6] => ? = 6 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [4,3,2,1,7,6,5] => ? = 6 - 1
[[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [6,7,5,8,1,2,3,4] => [4,3,2,1,8,5,7,6] => [4,3,2,1,5,7,6] => ? = 5 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [7,5,6,8,1,2,3,4] => [4,3,2,1,8,6,5,7] => [4,3,2,1,6,5,7] => ? = 5 - 1
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,1,2,3,4] => [4,3,2,1,8,7,5,6] => [4,3,2,1,7,5,6] => ? = 6 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [4,3,2,1,7,6,5] => ? = 6 - 1
[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [8,7,6,4,3,1,2,5] => [5,2,1,3,4,6,7,8] => ? => ? = 5 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,1,2,5] => [5,2,1,3,4,8,6,7] => ? => ? = 5 - 1
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [6,7,8,4,3,1,2,5] => [5,2,1,3,4,8,7,6] => ? => ? = 6 - 1
[[[.,.],[[.,.],.]],[.,[.,[.,.]]]]
=> [8,7,6,3,4,1,2,5] => [5,2,1,4,3,6,7,8] => ? => ? = 5 - 1
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
=> [7,6,8,3,4,1,2,5] => [5,2,1,4,3,8,6,7] => ? => ? = 5 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [5,2,1,4,3,8,7,6] => [5,2,1,4,3,7,6] => ? = 6 - 1
[[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [8,7,6,4,1,2,3,5] => [5,3,2,1,4,6,7,8] => ? => ? = 5 - 1
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,1,2,3,5] => [5,3,2,1,4,8,6,7] => ? => ? = 5 - 1
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [6,7,8,4,1,2,3,5] => [5,3,2,1,4,8,7,6] => ? => ? = 6 - 1
[[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> [8,7,6,3,1,2,4,5] => [5,4,2,1,3,6,7,8] => ? => ? = 5 - 1
[[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => [5,4,2,1,3,8,6,7] => ? => ? = 5 - 1
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [6,7,8,3,1,2,4,5] => [5,4,2,1,3,8,7,6] => ? => ? = 6 - 1
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [7,6,4,3,1,2,5,8] => [8,5,2,1,3,4,6,7] => ? => ? = 5 - 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [6,7,3,4,1,2,5,8] => [8,5,2,1,4,3,7,6] => [5,2,1,4,3,7,6] => ? = 6 - 1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => [8,6,4,2,1,3,5,7] => [6,4,2,1,3,5,7] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [7,5,1,2,3,4,6,8] => [8,6,4,3,2,1,5,7] => [6,4,3,2,1,5,7] => ? = 6 - 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [7,3,1,2,4,5,6,8] => [8,6,5,4,2,1,3,7] => [6,5,4,2,1,3,7] => ? = 6 - 1
[[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [6,4,5,1,2,3,7,8] => [8,7,3,2,1,5,4,6] => ? => ? = 7 - 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [6,4,1,2,3,5,7,8] => [8,7,5,3,2,1,4,6] => ? => ? = 7 - 1
[[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [5,3,1,2,4,6,7,8] => [8,7,6,4,2,1,3,5] => [7,6,4,2,1,3,5] => ? = 7 - 1
[[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [5,1,2,3,4,6,7,8] => [8,7,6,4,3,2,1,5] => [7,6,4,3,2,1,5] => ? = 7 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [3,1,2,4,5,6,7,8] => [8,7,6,5,4,2,1,3] => [7,6,5,4,2,1,3] => ? = 7 - 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: St001300
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001300: Posets ⟶ ℤResult quality: 94% ●values known / values provided: 94%●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
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [6,7,8,5,4,1,2,3] => [6,7,5,4,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,1,2,3] => [5,6,7,4,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 5 - 1
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 4 - 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,1,2,3] => ? => ?
=> ? = 5 - 1
[[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => ? => ?
=> ? = 6 - 1
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [5,4,6,7,8,1,2,3] => ? => ?
=> ? = 6 - 1
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,1,2,3] => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 6 - 1
[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [8,7,6,5,3,1,2,4] => [7,6,5,3,1,2,4] => ([(3,6),(4,5),(5,6)],7)
=> ? = 4 - 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [8,6,7,5,1,2,3,4] => [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 5 - 1
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> [6,7,8,5,1,2,3,4] => [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 5 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 5 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 5 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 6 - 1
[[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [6,7,5,8,1,2,3,4] => [6,7,5,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 5 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 5 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 6 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,1,2,5] => ? => ?
=> ? = 5 - 1
[[[.,.],[[.,.],.]],[.,[.,[.,.]]]]
=> [8,7,6,3,4,1,2,5] => [7,6,3,4,1,2,5] => ([(2,5),(3,4),(4,6),(5,6)],7)
=> ? = 5 - 1
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
=> [7,6,8,3,4,1,2,5] => ? => ?
=> ? = 5 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 6 - 1
[[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [8,7,6,4,1,2,3,5] => [7,6,4,1,2,3,5] => ([(2,6),(3,4),(4,5),(5,6)],7)
=> ? = 5 - 1
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,1,2,3,5] => ? => ?
=> ? = 5 - 1
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [6,7,8,4,1,2,3,5] => [6,7,4,1,2,3,5] => ([(0,6),(1,3),(2,4),(4,5),(5,6)],7)
=> ? = 6 - 1
[[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> [8,7,6,3,1,2,4,5] => [7,6,3,1,2,4,5] => ([(2,6),(3,4),(4,6),(6,5)],7)
=> ? = 5 - 1
[[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => ? => ?
=> ? = 5 - 1
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [6,7,8,3,1,2,4,5] => [6,7,3,1,2,4,5] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 6 - 1
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 5 - 1
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 5 - 1
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 5 - 1
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 6 - 1
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 6 - 1
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 6 - 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 - 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [6,7,3,4,1,2,5,8] => ? => ?
=> ? = 6 - 1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => [7,5,3,1,2,4,6] => ([(1,6),(2,5),(3,4),(4,6),(6,5)],7)
=> ? = 6 - 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [7,3,1,2,4,5,6,8] => [7,3,1,2,4,5,6] => ([(1,6),(2,3),(3,6),(4,5),(6,4)],7)
=> ? = 6 - 1
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 6 - 1
[[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [6,4,5,1,2,3,7,8] => ? => ?
=> ? = 7 - 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [6,4,1,2,3,5,7,8] => [6,4,1,2,3,5,7] => ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 7 - 1
[[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [5,3,1,2,4,6,7,8] => [5,3,1,2,4,6,7] => ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 7 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7] => ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? = 7 - 1
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 7 - 1
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Matching statistic: St000147
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 93%●distinct values known / distinct values provided: 86%
Values
[.,[.,.]]
=> [1,1,0,0]
=> []
=> ?
=> ? = 1 - 1
[[.,.],.]
=> [1,0,1,0]
=> [1]
=> []
=> 0 = 1 - 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0 = 1 - 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> 0 = 1 - 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0 = 1 - 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0 = 1 - 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0 = 1 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2 = 3 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 2 = 3 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 2 = 3 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 2 = 3 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2 = 3 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2 = 3 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2 = 3 - 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2 = 3 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> 3 = 4 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 3 = 4 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 3 = 4 - 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 3 = 4 - 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> 0 = 1 - 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 1 = 2 - 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 2 = 3 - 1
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 2 = 3 - 1
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 1 - 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,2,2,1]
=> ?
=> ? = 5 - 1
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> [5,4,3,2,2,1]
=> ? = 6 - 1
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ?
=> ? = 4 - 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ?
=> ? = 4 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2,1]
=> ?
=> ? = 6 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2,1]
=> ?
=> ? = 6 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2,1]
=> [5,3,3,3,2,1]
=> ? = 6 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> ? = 6 - 1
[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,1,1,1]
=> ?
=> ? = 5 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,1,1,1]
=> ?
=> ? = 5 - 1
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1,1,1]
=> ?
=> ? = 6 - 1
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,2,1,1]
=> ?
=> ? = 5 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2,1,1]
=> ?
=> ? = 6 - 1
[[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,2,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,2,2,1]
=> ?
=> ? = 5 - 1
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2,2,1]
=> [5,4,4,2,2,1]
=> ? = 6 - 1
[[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,1,1]
=> ?
=> ? = 5 - 1
[[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,1,1]
=> ?
=> ? = 5 - 1
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,1,1]
=> ?
=> ? = 6 - 1
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> ? = 5 - 1
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> ? = 5 - 1
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> ? = 5 - 1
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> ? = 6 - 1
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? = 6 - 1
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? = 6 - 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,1,1,1]
=> ?
=> ? = 5 - 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,2,1,1]
=> ?
=> ? = 6 - 1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ?
=> ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,2,1]
=> ?
=> ? = 6 - 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,1,1]
=> ?
=> ? = 6 - 1
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> ? = 6 - 1
[[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,2,2,1]
=> ?
=> ? = 7 - 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2,1]
=> ?
=> ? = 7 - 1
[[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,1,1]
=> ?
=> ? = 7 - 1
[[[[[[[.,.],.],.],[.,.]],.],.],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ?
=> ? = 7 - 1
[[[[[[[.,.],[.,.]],.],.],.],.],.]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> ? = 7 - 1
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 7 - 1
Description
The largest part of an integer partition.
Matching statistic: St000141
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 92% ●values known / values provided: 92%●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
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,2,6,7,8,5,4,3] => [3,4,5,8,7,6,2,1] => [3,4,5,7,6,2,1] => ? = 4 - 1
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> [1,2,5,6,7,8,4,3] => [3,4,8,7,6,5,2,1] => [3,4,7,6,5,2,1] => ? = 5 - 1
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,2,4,8,7,6,5,3] => [3,5,6,7,8,4,2,1] => [3,5,6,7,4,2,1] => ? = 4 - 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,2,4,6,7,8,5,3] => [3,5,8,7,6,4,2,1] => [3,5,7,6,4,2,1] => ? = 5 - 1
[[[.,.],.],[[[.,[.,[.,.]]],.],.]]
=> [1,2,6,5,4,7,8,3] => [3,8,7,4,5,6,2,1] => ? => ? = 6 - 1
[[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [1,2,5,6,4,7,8,3] => [3,8,7,4,6,5,2,1] => ? => ? = 6 - 1
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [1,2,5,4,6,7,8,3] => [3,8,7,6,4,5,2,1] => [3,7,6,4,5,2,1] => ? = 6 - 1
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,2,4,5,6,7,8,3] => [3,8,7,6,5,4,2,1] => ? => ? = 6 - 1
[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,3,2,8,7,6,5,4] => [4,5,6,7,8,2,3,1] => [4,5,6,7,2,3,1] => ? = 4 - 1
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,8,7,6,5,4] => [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => ? = 4 - 1
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,2,3,7,8,6,5,4] => [4,5,6,8,7,3,2,1] => ? => ? = 4 - 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,2,3,6,8,7,5,4] => [4,5,7,8,6,3,2,1] => ? => ? = 5 - 1
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,2,3,7,6,8,5,4] => [4,5,8,6,7,3,2,1] => ? => ? = 4 - 1
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> [1,2,3,6,7,8,5,4] => [4,5,8,7,6,3,2,1] => ? => ? = 5 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => [4,6,7,8,5,3,2,1] => ? => ? = 5 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,2,3,5,7,8,6,4] => [4,6,8,7,5,3,2,1] => ? => ? = 5 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [1,2,3,6,5,8,7,4] => [4,7,8,5,6,3,2,1] => ? => ? = 6 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,2,3,5,6,8,7,4] => [4,7,8,6,5,3,2,1] => ? => ? = 6 - 1
[[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,2,3,7,6,5,8,4] => [4,8,5,6,7,3,2,1] => ? => ? = 4 - 1
[[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [1,2,3,6,7,5,8,4] => [4,8,5,7,6,3,2,1] => ? => ? = 5 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,2,3,5,7,6,8,4] => [4,8,6,7,5,3,2,1] => ? => ? = 5 - 1
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,2,3,6,5,7,8,4] => [4,8,7,5,6,3,2,1] => [4,7,5,6,3,2,1] => ? = 6 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,5,6,7,8,4] => [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => ? = 6 - 1
[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,4,3,2,8,7,6,5] => [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => ? = 5 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [1,4,3,2,7,6,8,5] => [5,8,6,7,2,3,4,1] => ? => ? = 5 - 1
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [1,4,3,2,6,7,8,5] => [5,8,7,6,2,3,4,1] => ? => ? = 6 - 1
[[[.,.],[[.,.],.]],[.,[.,[.,.]]]]
=> [1,3,4,2,8,7,6,5] => [5,6,7,8,2,4,3,1] => [5,6,7,2,4,3,1] => ? = 5 - 1
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
=> [1,3,4,2,7,6,8,5] => [5,8,6,7,2,4,3,1] => ? => ? = 5 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [1,3,4,2,6,7,8,5] => [5,8,7,6,2,4,3,1] => [5,7,6,2,4,3,1] => ? = 6 - 1
[[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,2,4,3,8,7,6,5] => [5,6,7,8,3,4,2,1] => [5,6,7,3,4,2,1] => ? = 5 - 1
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [1,2,4,3,7,6,8,5] => [5,8,6,7,3,4,2,1] => ? => ? = 5 - 1
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [1,2,4,3,6,7,8,5] => [5,8,7,6,3,4,2,1] => [5,7,6,3,4,2,1] => ? = 6 - 1
[[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> [1,3,2,4,8,7,6,5] => [5,6,7,8,4,2,3,1] => [5,6,7,4,2,3,1] => ? = 5 - 1
[[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [1,3,2,4,7,6,8,5] => [5,8,6,7,4,2,3,1] => ? => ? = 5 - 1
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,3,2,4,6,7,8,5] => [5,8,7,6,4,2,3,1] => [5,7,6,4,2,3,1] => ? = 6 - 1
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,8,7,6,5] => [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 5 - 1
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,2,3,4,7,8,6,5] => [5,6,8,7,4,3,2,1] => ? => ? = 5 - 1
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,2,3,4,7,6,8,5] => [5,8,6,7,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 5 - 1
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,6,7,8,5] => [5,8,7,6,4,3,2,1] => [5,7,6,4,3,2,1] => ? = 6 - 1
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,5,8,7,6] => [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 6 - 1
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,7,8,6] => [6,8,7,5,4,3,2,1] => ? => ? = 6 - 1
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => ? = 5 - 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,3,4,2,6,7,5,8] => [8,5,7,6,2,4,3,1] => [5,7,6,2,4,3,1] => ? = 6 - 1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => [8,6,7,4,5,2,3,1] => [6,7,4,5,2,3,1] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,2,3,5,4,7,6,8] => [8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => ? = 6 - 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => ? = 6 - 1
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => [8,6,7,5,4,3,2,1] => ? => ? = 6 - 1
[[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [1,2,4,6,5,3,7,8] => [8,7,3,5,6,4,2,1] => ? => ? = 7 - 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => ? = 7 - 1
[[[[[[.,.],[.,.]],[.,.]],.],.],.]
=> [1,3,2,5,4,6,7,8] => [8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => ? = 7 - 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: St001291
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: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
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
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [1,2,6,7,8,5,4,3] => [1,2,6,7,5,4,3] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 4
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> [1,2,5,6,7,8,4,3] => [1,2,5,6,7,4,3] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 5
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [1,2,4,8,7,6,5,3] => [1,2,4,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 4
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [1,2,4,6,7,8,5,3] => [1,2,4,6,7,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 5
[[[.,.],.],[[[.,[.,[.,.]]],.],.]]
=> [1,2,6,5,4,7,8,3] => [1,2,6,5,4,7,3] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [1,2,5,6,4,7,8,3] => [1,2,5,6,4,7,3] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 6
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [1,2,5,4,6,7,8,3] => [1,2,5,4,6,7,3] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 6
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,3] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [1,3,2,8,7,6,5,4] => [1,3,2,7,6,5,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [1,2,3,8,7,6,5,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [1,2,3,7,8,6,5,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [1,2,3,6,8,7,5,4] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [1,2,3,7,6,8,5,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> [1,2,3,6,7,8,5,4] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [1,2,3,5,8,7,6,4] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 5
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [1,2,3,5,7,8,6,4] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 5
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [1,2,3,6,5,8,7,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 6
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [1,2,3,5,6,8,7,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [1,2,3,7,6,5,8,4] => [1,2,3,7,6,5,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [1,2,3,6,7,5,8,4] => [1,2,3,6,7,5,4] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 5
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [1,2,3,5,7,6,8,4] => [1,2,3,5,7,6,4] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 5
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [1,2,3,6,5,7,8,4] => [1,2,3,6,5,7,4] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 6
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [1,4,3,2,8,7,6,5] => [1,4,3,2,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [1,4,3,2,7,6,8,5] => [1,4,3,2,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [1,4,3,2,6,7,8,5] => [1,4,3,2,6,7,5] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 6
[[[.,.],[[.,.],.]],[.,[.,[.,.]]]]
=> [1,3,4,2,8,7,6,5] => [1,3,4,2,7,6,5] => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
=> [1,3,4,2,7,6,8,5] => [1,3,4,2,7,6,5] => [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 6
[[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [1,2,4,3,8,7,6,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [1,2,4,3,7,6,8,5] => [1,2,4,3,7,6,5] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [1,2,4,3,6,7,8,5] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 6
[[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> [1,3,2,4,8,7,6,5] => [1,3,2,4,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [1,3,2,4,7,6,8,5] => [1,3,2,4,7,6,5] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 5
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [1,3,2,4,6,7,8,5] => [1,3,2,4,6,7,5] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 6
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [1,2,3,4,8,7,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [1,2,3,4,7,8,6,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [1,2,3,4,7,6,8,5] => [1,2,3,4,7,6,5] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 5
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [1,2,3,4,5,8,7,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [1,2,3,4,5,7,8,6] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 7
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 5
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [1,3,4,2,6,7,5,8] => [1,3,4,2,6,7,5] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 6
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 6
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 6
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 6
[[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [1,2,4,6,5,3,7,8] => [1,2,4,6,5,3,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 7
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 7
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: St000316
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 92%●distinct values known / distinct values provided: 86%
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
[[[.,.],.],[.,[.,[[[.,.],.],.]]]]
=> [6,7,8,5,4,1,2,3] => [3,2,1,4,5,8,7,6] => [3,2,1,4,5,7,6] => ? = 4 - 1
[[[.,.],.],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,1,2,3] => [3,2,1,4,8,7,6,5] => [3,2,1,4,7,6,5] => ? = 5 - 1
[[[.,.],.],[[.,.],[.,[.,[.,.]]]]]
=> [8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? => ? = 4 - 1
[[[.,.],.],[[.,.],[[[.,.],.],.]]]
=> [6,7,8,4,5,1,2,3] => [3,2,1,5,4,8,7,6] => ? => ? = 5 - 1
[[[.,.],.],[[[.,[.,[.,.]]],.],.]]
=> [6,5,4,7,8,1,2,3] => [3,2,1,8,7,4,5,6] => [3,2,1,7,4,5,6] => ? = 6 - 1
[[[.,.],.],[[[.,[[.,.],.]],.],.]]
=> [5,6,4,7,8,1,2,3] => [3,2,1,8,7,4,6,5] => ? => ? = 6 - 1
[[[.,.],.],[[[[.,[.,.]],.],.],.]]
=> [5,4,6,7,8,1,2,3] => [3,2,1,8,7,6,4,5] => ? => ? = 6 - 1
[[[.,.],.],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,1,2,3] => [3,2,1,8,7,6,5,4] => [3,2,1,7,6,5,4] => ? = 6 - 1
[[[.,.],[.,.]],[.,[.,[.,[.,.]]]]]
=> [8,7,6,5,3,1,2,4] => [4,2,1,3,5,6,7,8] => ? => ? = 4 - 1
[[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
=> [8,7,6,5,1,2,3,4] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[.,[.,[[.,.],.]]]]
=> [7,8,6,5,1,2,3,4] => [4,3,2,1,5,6,8,7] => [4,3,2,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[.,[[.,.],[.,.]]]]
=> [8,6,7,5,1,2,3,4] => [4,3,2,1,5,7,6,8] => [4,3,2,1,5,7,6] => ? = 5 - 1
[[[[.,.],.],.],[.,[[.,[.,.]],.]]]
=> [7,6,8,5,1,2,3,4] => [4,3,2,1,5,8,6,7] => [4,3,2,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[.,[[[.,.],.],.]]]
=> [6,7,8,5,1,2,3,4] => [4,3,2,1,5,8,7,6] => [4,3,2,1,5,7,6] => ? = 5 - 1
[[[[.,.],.],.],[[.,.],[.,[.,.]]]]
=> [8,7,5,6,1,2,3,4] => [4,3,2,1,6,5,7,8] => [4,3,2,1,6,5,7] => ? = 5 - 1
[[[[.,.],.],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,1,2,3,4] => [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,7] => ? = 5 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [8,6,5,7,1,2,3,4] => [4,3,2,1,7,5,6,8] => [4,3,2,1,7,5,6] => ? = 6 - 1
[[[[.,.],.],.],[[[.,.],.],[.,.]]]
=> [8,5,6,7,1,2,3,4] => [4,3,2,1,7,6,5,8] => [4,3,2,1,7,6,5] => ? = 6 - 1
[[[[.,.],.],.],[[.,[.,[.,.]]],.]]
=> [7,6,5,8,1,2,3,4] => [4,3,2,1,8,5,6,7] => [4,3,2,1,5,6,7] => ? = 4 - 1
[[[[.,.],.],.],[[.,[[.,.],.]],.]]
=> [6,7,5,8,1,2,3,4] => [4,3,2,1,8,5,7,6] => [4,3,2,1,5,7,6] => ? = 5 - 1
[[[[.,.],.],.],[[[.,.],[.,.]],.]]
=> [7,5,6,8,1,2,3,4] => [4,3,2,1,8,6,5,7] => [4,3,2,1,6,5,7] => ? = 5 - 1
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,1,2,3,4] => [4,3,2,1,8,7,5,6] => [4,3,2,1,7,5,6] => ? = 6 - 1
[[[[.,.],.],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,1,2,3,4] => [4,3,2,1,8,7,6,5] => [4,3,2,1,7,6,5] => ? = 6 - 1
[[[.,.],[.,[.,.]]],[.,[.,[.,.]]]]
=> [8,7,6,4,3,1,2,5] => [5,2,1,3,4,6,7,8] => ? => ? = 5 - 1
[[[.,.],[.,[.,.]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,1,2,5] => [5,2,1,3,4,8,6,7] => ? => ? = 5 - 1
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [6,7,8,4,3,1,2,5] => [5,2,1,3,4,8,7,6] => ? => ? = 6 - 1
[[[.,.],[[.,.],.]],[.,[.,[.,.]]]]
=> [8,7,6,3,4,1,2,5] => [5,2,1,4,3,6,7,8] => ? => ? = 5 - 1
[[[.,.],[[.,.],.]],[[.,[.,.]],.]]
=> [7,6,8,3,4,1,2,5] => [5,2,1,4,3,8,6,7] => ? => ? = 5 - 1
[[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> [6,7,8,3,4,1,2,5] => [5,2,1,4,3,8,7,6] => [5,2,1,4,3,7,6] => ? = 6 - 1
[[[[.,.],.],[.,.]],[.,[.,[.,.]]]]
=> [8,7,6,4,1,2,3,5] => [5,3,2,1,4,6,7,8] => ? => ? = 5 - 1
[[[[.,.],.],[.,.]],[[.,[.,.]],.]]
=> [7,6,8,4,1,2,3,5] => [5,3,2,1,4,8,6,7] => ? => ? = 5 - 1
[[[[.,.],.],[.,.]],[[[.,.],.],.]]
=> [6,7,8,4,1,2,3,5] => [5,3,2,1,4,8,7,6] => ? => ? = 6 - 1
[[[[.,.],[.,.]],.],[.,[.,[.,.]]]]
=> [8,7,6,3,1,2,4,5] => [5,4,2,1,3,6,7,8] => ? => ? = 5 - 1
[[[[.,.],[.,.]],.],[[.,[.,.]],.]]
=> [7,6,8,3,1,2,4,5] => [5,4,2,1,3,8,6,7] => ? => ? = 5 - 1
[[[[.,.],[.,.]],.],[[[.,.],.],.]]
=> [6,7,8,3,1,2,4,5] => [5,4,2,1,3,8,7,6] => ? => ? = 6 - 1
[[[[[.,.],.],.],.],[.,[.,[.,.]]]]
=> [8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7] => ? = 5 - 1
[[[[[.,.],.],.],.],[.,[[.,.],.]]]
=> [7,8,6,1,2,3,4,5] => [5,4,3,2,1,6,8,7] => [5,4,3,2,1,6,7] => ? = 5 - 1
[[[[[.,.],.],.],.],[[.,[.,.]],.]]
=> [7,6,8,1,2,3,4,5] => [5,4,3,2,1,8,6,7] => [5,4,3,2,1,6,7] => ? = 5 - 1
[[[[[.,.],.],.],.],[[[.,.],.],.]]
=> [6,7,8,1,2,3,4,5] => [5,4,3,2,1,8,7,6] => [5,4,3,2,1,7,6] => ? = 6 - 1
[[[[[[.,.],.],.],.],.],[.,[.,.]]]
=> [8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7] => ? = 6 - 1
[[[[[[.,.],.],.],.],.],[[.,.],.]]
=> [7,8,1,2,3,4,5,6] => [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,7] => ? = 6 - 1
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ? = 7 - 1
[[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
=> [7,6,4,3,1,2,5,8] => [8,5,2,1,3,4,6,7] => ? => ? = 5 - 1
[[[[.,.],[[.,.],.]],[[.,.],.]],.]
=> [6,7,3,4,1,2,5,8] => [8,5,2,1,4,3,7,6] => [5,2,1,4,3,7,6] => ? = 6 - 1
[[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> [7,5,3,1,2,4,6,8] => [8,6,4,2,1,3,5,7] => [6,4,2,1,3,5,7] => ? = 6 - 1
[[[[[[.,.],.],.],[.,.]],[.,.]],.]
=> [7,5,1,2,3,4,6,8] => [8,6,4,3,2,1,5,7] => [6,4,3,2,1,5,7] => ? = 6 - 1
[[[[[[.,.],[.,.]],.],.],[.,.]],.]
=> [7,3,1,2,4,5,6,8] => [8,6,5,4,2,1,3,7] => [6,5,4,2,1,3,7] => ? = 6 - 1
[[[[[[[.,.],.],.],.],.],[.,.]],.]
=> [7,1,2,3,4,5,6,8] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 6 - 1
[[[[[.,.],.],[[.,.],[.,.]]],.],.]
=> [6,4,5,1,2,3,7,8] => [8,7,3,2,1,5,4,6] => ? => ? = 7 - 1
[[[[[[.,.],.],[.,.]],[.,.]],.],.]
=> [6,4,1,2,3,5,7,8] => [8,7,5,3,2,1,4,6] => ? => ? = 7 - 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 $\sigma_j > \sigma_i$.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000653The last descent of a permutation. 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.
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