Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000141
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00064: Permutations reversePermutations
Mp00252: Permutations restrictionPermutations
St000141: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1] => 0
[[.,.],.]
=> [1,2] => [2,1] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 0
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,3,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [2,3,1] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [2,3,1] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [3,1,2] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [3,2,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [2,3,1] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,2] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,3,4,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,4,3,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,2] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,1] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,4,1,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,4,1,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [4,1,2,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,1,3,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 3
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,3,1,2] => 3
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0
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: St000734
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00252: Permutations restrictionPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 95% values known / values provided: 95%distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [[1]]
=> 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => [[1]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [[1],[2]]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [[1,2]]
=> 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [[1,3],[2]]
=> 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [[1],[2],[3]]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [[1,2],[3]]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [[1,2],[3]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [[1,3],[2]]
=> 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [[1,2,3]]
=> 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [6,8,7,5,4,3,2,1] => ? => ?
=> ? = 1 + 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [2,3,4,6,7,8,5,1] => ? => ?
=> ? = 4 + 1
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [2,3,4,5,8,7,6,1] => ? => ?
=> ? = 4 + 1
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,7,8,6,1] => ? => ?
=> ? = 4 + 1
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [6,5,4,3,2,8,7,1] => ? => ?
=> ? = 5 + 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [2,3,5,6,7,4,8,1] => ? => ?
=> ? = 4 + 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [2,3,4,5,7,6,8,1] => ? => ?
=> ? = 4 + 1
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [2,3,6,5,4,7,8,1] => ? => ?
=> ? = 5 + 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [2,3,5,6,4,7,8,1] => ? => ?
=> ? = 5 + 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? => ?
=> ? = 1 + 1
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [3,2,1,6,5,4,8,7] => ? => ?
=> ? = 6 + 1
[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> [6,5,4,3,2,7,1,8] => ? => ?
=> ? = 5 + 1
[[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [2,1,7,6,5,4,3,8] => ? => ?
=> ? = 2 + 1
[[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [3,2,1,6,5,4,7,8] => ? => ?
=> ? = 6 + 1
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [2,1,4,3,6,5,7,8] => ? => ?
=> ? = 6 + 1
[[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [3,2,1,5,4,6,7,8] => ? => ?
=> ? = 6 + 1
[[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [2,1,4,3,5,6,7,8] => ? => ?
=> ? = 6 + 1
[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> [7,9,8,6,5,4,3,2,1] => ? => ?
=> ? = 1 + 1
[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [2,1,9,8,7,6,5,4,3] => ? => ?
=> ? = 2 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
=> [5,4,3,2,1,9,8,7,6] => ? => ?
=> ? = 5 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,.]]]
=> [6,5,4,3,2,1,9,8,7] => ? => ?
=> ? = 6 + 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]
=> [7,6,5,4,3,2,1,9,8] => ? => ?
=> ? = 7 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1,10,9,8,7,6] => ? => ?
=> ? = 5 + 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> [7,6,5,4,3,2,1,10,9,8] => ? => ?
=> ? = 7 + 1
[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> [4,9,8,7,6,5,3,2,1] => ? => ?
=> ? = 1 + 1
[.,[[[[[[[.,.],.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,7,9,8,1] => ? => ?
=> ? = 6 + 1
[.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,6,8,9,7,1] => ? => ?
=> ? = 5 + 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]
=> [2,3,4,5,6,7,8,10,9,1] => ? => ?
=> ? = 7 + 1
[[[[[[[.,[.,.]],[.,.]],.],.],.],.],.]
=> [2,1,4,3,5,6,7,8,9] => ? => ?
=> ? = 7 + 1
[[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]],.]
=> [6,5,4,3,2,1,8,7,9] => ? => ?
=> ? = 6 + 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> [7,6,5,4,3,2,9,8,1] => ? => ?
=> ? = 6 + 1
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> [8,7,6,5,4,3,2,10,9,1] => ? => ?
=> ? = 7 + 1
[.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]]
=> [10,11,9,8,7,6,5,4,3,2,1] => ? => ?
=> ? = 0 + 1
[[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]],.]
=> [7,6,5,4,3,2,8,1,9] => ? => ?
=> ? = 6 + 1
[[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]
=> [5,4,3,2,1,8,7,6,9] => ? => ?
=> ? = 5 + 1
Description
The last entry in the first row of a standard tableau.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00064: Permutations reversePermutations
Mp00252: Permutations restrictionPermutations
St000019: Permutations ⟶ ℤResult quality: 94% values known / values provided: 94%distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1] => 0
[[.,.],.]
=> [1,2] => [2,1] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,3,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,3] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [2,1,3] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5] => ? = 4
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,4,3,2,7,6,5] => ? = 4
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [8,7,6,4,3,2,5,1] => [1,5,2,3,4,6,7,8] => [1,5,2,3,4,6,7] => ? = 3
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => ? => ? = 3
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => ? => ? = 4
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => [1,6,2,3,4,5,7,8] => [1,6,2,3,4,5,7] => ? = 4
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [7,8,5,4,3,2,6,1] => [1,6,2,3,4,5,8,7] => ? => ? = 4
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [8,7,2,3,4,5,6,1] => [1,6,5,4,3,2,7,8] => ? => ? = 4
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => [1,6,5,4,3,2,8,7] => ? => ? = 4
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => [1,7,6,5,4,3,2,8] => ? => ? = 5
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [5,6,7,2,3,4,8,1] => [1,8,4,3,2,7,6,5] => ? => ? = 4
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [7,6,4,3,2,5,8,1] => [1,8,5,2,3,4,6,7] => ? => ? = 3
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => [1,8,6,2,3,4,5,7] => ? => ? = 4
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [7,2,3,4,5,6,8,1] => [1,8,6,5,4,3,2,7] => ? => ? = 4
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [6,5,2,3,4,7,8,1] => [1,8,7,4,3,2,5,6] => [1,7,4,3,2,5,6] => ? = 5
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => [1,8,7,4,3,2,6,5] => ? => ? = 5
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,2,1,5] => [5,1,2,3,4,8,6,7] => ? => ? = 4
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [8,6,5,3,2,1,4,7] => [7,4,1,2,3,5,6,8] => ? => ? = 6
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [8,6,4,2,1,3,5,7] => [7,5,3,1,2,4,6,8] => [7,5,3,1,2,4,6] => ? = 6
[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [8,5,4,3,2,1,6,7] => [7,6,1,2,3,4,5,8] => ? => ? = 6
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [7,5,4,3,2,6,1,8] => [8,1,6,2,3,4,5,7] => ? => ? = 4
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [7,2,3,4,5,6,1,8] => [8,1,6,5,4,3,2,7] => ? => ? = 4
[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [7,6,4,3,2,1,5,8] => [8,5,1,2,3,4,6,7] => ? => ? = 4
[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [7,5,4,3,2,1,6,8] => [8,6,1,2,3,4,5,7] => ? => ? = 5
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [7,5,3,2,1,4,6,8] => [8,6,4,1,2,3,5,7] => ? => ? = 5
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [6,4,2,1,3,5,7,8] => [8,7,5,3,1,2,4,6] => ? => ? = 6
[[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [5,3,2,1,4,6,7,8] => [8,7,6,4,1,2,3,5] => [7,6,4,1,2,3,5] => ? = 6
[[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [4,2,1,3,5,6,7,8] => [8,7,6,5,3,1,2,4] => [7,6,5,3,1,2,4] => ? = 6
[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> [9,7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7,9] => ? => ? = 1
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [10,8,6,4,2,1,3,5,7,9] => [9,7,5,3,1,2,4,6,8,10] => ? => ? = 8
[[.,[.,.]],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [9,8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8,9] => ? => ? = 2
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,.]]]]
=> [9,8,7,5,4,3,2,1,6] => [6,1,2,3,4,5,7,8,9] => ? => ? = 5
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> [10,9,8,7,5,4,3,2,1,6] => [6,1,2,3,4,5,7,8,9,10] => ? => ? = 5
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> [10,9,8,6,5,4,3,2,1,7] => [7,1,2,3,4,5,6,8,9,10] => ? => ? = 6
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> [10,9,7,6,5,4,3,2,1,8] => [8,1,2,3,4,5,6,7,9,10] => ? => ? = 7
[.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> [9,8,7,6,4,5,3,2,1] => [1,2,3,5,4,6,7,8,9] => ? => ? = 1
[.,[[[[[[[.,.],.],.],.],.],.],[.,.]]]
=> [9,2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2,9] => ? => ? = 6
[.,[[[[[[.,.],.],.],.],.],[[.,.],.]]]
=> [8,9,2,3,4,5,6,7,1] => [1,7,6,5,4,3,2,9,8] => ? => ? = 5
[.,[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]]
=> [10,2,3,4,5,6,7,8,9,1] => [1,9,8,7,6,5,4,3,2,10] => ? => ? = 7
[[[[[[[.,[.,.]],[.,.]],.],.],.],.],.]
=> [4,2,1,3,5,6,7,8,9] => [9,8,7,6,5,3,1,2,4] => ? => ? = 7
[[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]],.]
=> [8,6,5,4,3,2,1,7,9] => [9,7,1,2,3,4,5,6,8] => ? => ? = 6
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> [9,7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7,9] => ? => ? = 6
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,.]]]
=> [10,8,7,6,5,4,3,2,9,1] => [1,9,2,3,4,5,6,7,8,10] => ? => ? = 7
[[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]],.]
=> [8,7,5,4,3,2,1,6,9] => [9,6,1,2,3,4,5,7,8] => ? => ? = 5
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: St000147
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00027: Dyck paths to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 92% values known / values provided: 92%distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> []
=> ?
=> ? = 0
[[.,.],.]
=> [1,0,1,0]
=> [1]
=> []
=> 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> []
=> ?
=> ? = 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [2]
=> []
=> 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 3
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 3
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 3
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> 1
[[.,[[.,[.,.]],.]],.]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> 2
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> 2
[.,[.,[.,[.,[.,[.,.]]]]]]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4]
=> ?
=> ? = 4
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3]
=> ?
=> ? = 6
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4]
=> ?
=> ? = 6
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2]
=> ?
=> ? = 6
[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4]
=> ?
=> ? = 6
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2]
=> [6,5,4,3,2]
=> ? = 6
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [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]
=> ? = 6
[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4]
=> ?
=> ? = 4
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3]
=> ?
=> ? = 5
[[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3]
=> ?
=> ? = 6
[[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4]
=> ?
=> ? = 6
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2]
=> ?
=> ? = 6
[[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3]
=> ?
=> ? = 6
[[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2]
=> ?
=> ? = 6
[[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> [6,5,4,3]
=> ? = 6
[[[[[[[.,[.,.]],.],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2]
=> [6,5,4,3,2]
=> ? = 6
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [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]
=> ? = 6
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 6
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 7
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> ?
=> ? = 0
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 7
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ? = 8
[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ?
=> ?
=> ?
=> ? = 0
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ?
=> ?
=> ?
=> ? = 0
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> ?
=> ?
=> ?
=> ? = 0
[[[[[.,[.,.]],[.,.]],[.,.]],[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [8,8,6,6,4,4,2,2]
=> ?
=> ? = 8
[[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ? = 7
[[[[[[[[[.,.],.],.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ? = 8
[.,[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]]
=> ?
=> ?
=> ?
=> ? = 8
[[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ?
=> ?
=> ?
=> ? = 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,[.,[.,.]]]]]
=> ?
=> ?
=> ?
=> ? = 5
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,[.,[.,.]]]]
=> ?
=> ?
=> ?
=> ? = 6
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],[.,[.,.]]]
=> ?
=> ?
=> ?
=> ? = 7
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
=> ?
=> ?
=> ?
=> ? = 8
[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5]
=> [7,6,5]
=> ? = 7
[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4]
=> [7,6,5,4]
=> ? = 7
[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3]
=> [7,6,5,4,3]
=> ? = 7
[[[[[[[[.,[.,.]],.],.],.],.],.],.],.]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2]
=> [7,6,5,4,3,2]
=> ? = 7
[[[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6]
=> [8,7,6]
=> ? = 8
[[[[[[.,[.,[.,[.,[.,.]]]]],.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5]
=> [8,7,6,5]
=> ? = 8
[[[[[[[.,[.,[.,[.,.]]]],.],.],.],.],.],.]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4]
=> [8,7,6,5,4]
=> ? = 8
[[[[[[[[.,[.,[.,.]]],.],.],.],.],.],.],.]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3]
=> [8,7,6,5,4,3]
=> ? = 8
Description
The largest part of an integer partition.
Matching statistic: St001300
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00252: Permutations restrictionPermutations
Mp00065: Permutations permutation posetPosets
St001300: Posets ⟶ ℤResult quality: 78% values known / values provided: 88%distinct values known / distinct values provided: 78%
Values
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> 0
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 0
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => ? => ?
=> ? = 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 4
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 4
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [8,7,6,4,3,2,5,1] => ? => ?
=> ? = 3
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => ? => ?
=> ? = 3
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => ? => ?
=> ? = 4
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => ? => ?
=> ? = 4
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [7,8,5,4,3,2,6,1] => ? => ?
=> ? = 4
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [8,7,2,3,4,5,6,1] => ? => ?
=> ? = 4
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => ? => ?
=> ? = 4
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [8,6,5,4,3,2,7,1] => ? => ?
=> ? = 5
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => ? => ?
=> ? = 5
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [5,6,7,2,3,4,8,1] => ? => ?
=> ? = 4
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [7,6,4,3,2,5,8,1] => ? => ?
=> ? = 3
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => ? => ?
=> ? = 4
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [7,2,3,4,5,6,8,1] => ? => ?
=> ? = 4
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [6,5,2,3,4,7,8,1] => ? => ?
=> ? = 5
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => ? => ?
=> ? = 5
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 5
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,3,1,2] => ? => ?
=> ? = 1
[[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [7,8,6,4,3,2,1,5] => ? => ?
=> ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,2,1,5] => ? => ?
=> ? = 4
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [8,7,5,4,3,2,1,6] => ? => ?
=> ? = 5
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [8,6,5,3,2,1,4,7] => ? => ?
=> ? = 6
[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [8,5,4,3,2,1,6,7] => ? => ?
=> ? = 6
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [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)
=> ? = 6
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [7,5,4,3,2,6,1,8] => ? => ?
=> ? = 4
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [7,2,3,4,5,6,1,8] => ? => ?
=> ? = 4
[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> [6,5,4,3,2,7,1,8] => ? => ?
=> ? = 5
[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [7,5,4,3,2,1,6,8] => ? => ?
=> ? = 5
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [7,5,3,2,1,4,6,8] => ? => ?
=> ? = 5
[[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [6,5,3,2,1,4,7,8] => ? => ?
=> ? = 6
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [6,4,2,1,3,5,7,8] => ? => ?
=> ? = 6
[[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [5,3,2,1,4,6,7,8] => ? => ?
=> ? = 6
[[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [4,2,1,3,5,6,7,8] => ? => ?
=> ? = 6
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [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)
=> ? = 6
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [9,8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [8,9,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 0
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> [7,8,9,6,5,4,3,2,1] => [7,8,6,5,4,3,2,1] => ([(6,7)],8)
=> ? = 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,1] => [2,3,4,5,6,7,8,1] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 6
[[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 7
[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> [10,9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> [9,10,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 0
[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> [9,7,8,6,5,4,3,2,1] => ? => ?
=> ? = 1
[.,[[[[[[[[[.,.],.],.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,9,10,1] => [2,3,4,5,6,7,8,9,1] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
=> ? = 7
[[[[[[[[[[.,.],.],.],.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 8
[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> [8,7,9,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 0
[.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> [8,7,6,5,4,3,2,9,1] => [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 0
[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 0
[.,[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]]
=> [9,8,7,6,5,4,3,2,10,1] => [9,8,7,6,5,4,3,2,1] => ([],9)
=> ? = 0
Description
The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset.
Mp00020: Binary trees to Tamari-corresponding Dyck pathDyck paths
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00252: Permutations restrictionPermutations
St000653: Permutations ⟶ ℤResult quality: 78% values known / values provided: 86%distinct values known / distinct values provided: 78%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,2] => [1] => ? = 0
[[.,.],.]
=> [1,0,1,0]
=> [2,1] => [1] => ? = 0
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 0
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 0
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 2
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 2
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 2
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 2
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 3
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => 3
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,4] => 0
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ? = 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? => ? = 0
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,7,1,2,5,8] => [3,4,6,7,1,2,5] => ? = 4
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,6,7,8,1,2,5] => ? => ? = 4
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,2,5,3,4,6,7,8] => ? => ? = 3
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,1,2,6,7,8] => ? => ? = 3
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [3,4,5,7,8,1,2,6] => ? => ? = 4
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0]
=> [1,2,3,6,8,4,5,7] => ? => ? = 4
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2,7,8] => ? => ? = 4
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [3,4,5,6,8,1,2,7] => ? => ? = 4
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,2,3,4,7,5,6,8] => ? => ? = 5
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [3,4,5,6,7,1,2,8] => ? => ? = 5
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,8,5,6] => [7,1,2,3,4,5,6] => ? = 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,6,7,1,8,2,5] => [3,4,6,7,1,2,5] => ? = 4
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4,8,6,7] => ? => ? = 3
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [3,4,5,6,1,8,2,7] => ? => ? = 4
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [3,4,1,2,7,8,5,6] => [3,4,1,2,7,5,6] => ? = 5
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,6,1,7,8,2,5] => [3,4,6,1,7,2,5] => ? = 5
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => ? = 5
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => ? = 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [2,1,3,4,5,8,6,7] => ? => ? = 1
[[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [1,1,1,1,0,0,0,0,1,1,1,0,1,0,0,0]
=> [1,2,3,5,8,4,6,7] => ? => ? = 4
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,2,3,4,6,5,7,8] => ? => ? = 5
[[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [1,2,3,4,6,8,5,7] => ? => ? = 5
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5,7,6,8] => ? => ? = 6
[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,4,6,7,5,8] => ? => ? = 6
[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,6,7,4,8] => ? => ? = 6
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,6,7,2,8] => ? => ? = 6
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => [2,3,4,5,6,7,1] => ? = 6
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => [7,1,2,3,4,5,6] => ? = 1
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,8,7] => ? => ? = 4
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [3,4,5,6,1,2,8,7] => ? => ? = 4
[[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,3,2,4,5,6,8,7] => ? => ? = 2
[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,2,3,5,4,6,8,7] => ? => ? = 4
[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,8,7] => ? => ? = 5
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,8,7] => ? => ? = 5
[[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,3,5,7,8,6] => ? => ? = 6
[[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,7,8,6] => ? => ? = 6
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,7,8,6] => ? => ? = 6
[[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,5,6,7,8,4] => ? => ? = 6
[[[[[[[[.,.],.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ? = 6
[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8] => ? = 0
[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 0
[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7] => [8,1,2,3,4,5,6,7] => ? = 1
[.,[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,8,9,1,2] => [3,4,5,6,7,8,1,2] => ? = 6
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 $\pi(0) = n+1$.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00252: Permutations restrictionPermutations
Mp00069: Permutations complementPermutations
St000957: Permutations ⟶ ℤResult quality: 67% values known / values provided: 86%distinct values known / distinct values provided: 67%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 0
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [2,1,3,4] => 1
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1] => [1,3,2,4] => 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => ? => ? => ? = 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 4
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [5,6,7,2,3,4,1] => [3,2,1,6,5,4,7] => ? = 4
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [8,7,6,4,3,2,5,1] => ? => ? => ? = 3
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => ? => ? => ? = 3
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => ? => ? => ? = 4
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => ? => ? => ? = 4
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [7,8,5,4,3,2,6,1] => ? => ? => ? = 4
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [8,7,2,3,4,5,6,1] => ? => ? => ? = 4
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => ? => ? => ? = 4
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [8,6,5,4,3,2,7,1] => ? => ? => ? = 5
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => ? => ? => ? = 5
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [5,6,7,2,3,4,8,1] => ? => ? => ? = 4
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [7,6,4,3,2,5,8,1] => ? => ? => ? = 3
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => ? => ? => ? = 4
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [7,2,3,4,5,6,8,1] => ? => ? => ? = 4
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [6,5,2,3,4,7,8,1] => ? => ? => ? = 5
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => ? => ? => ? = 5
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [6,5,4,3,2,1,7] => ? = 5
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,3,1,2] => ? => ? => ? = 1
[[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [7,8,6,4,3,2,1,5] => ? => ? => ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,2,1,5] => ? => ? => ? = 4
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [8,7,5,4,3,2,1,6] => ? => ? => ? = 5
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [2,3,4,5,6,7,1] => ? = 6
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [8,6,5,3,2,1,4,7] => ? => ? => ? = 6
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [8,6,4,3,2,1,5,7] => [6,4,3,2,1,5,7] => [2,4,5,6,7,3,1] => ? = 6
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [8,6,4,2,1,3,5,7] => [6,4,2,1,3,5,7] => [2,4,6,7,5,3,1] => ? = 6
[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [8,5,4,3,2,1,6,7] => ? => ? => ? = 6
[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => ? = 6
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => ? = 6
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 6
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [6,7,5,4,3,2,1,8] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => ? = 1
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [7,5,4,3,2,6,1,8] => ? => ? => ? = 4
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [7,2,3,4,5,6,1,8] => ? => ? => ? = 4
[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> [6,5,4,3,2,7,1,8] => ? => ? => ? = 5
[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [7,5,4,3,2,1,6,8] => ? => ? => ? = 5
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [7,5,3,2,1,4,6,8] => ? => ? => ? = 5
[[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
=> [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7] => [2,3,4,5,6,7,1] => ? = 6
[[[[.,[.,[.,.]]],[.,[.,.]]],.],.]
=> [6,5,3,2,1,4,7,8] => ? => ? => ? = 6
[[[[.,[.,[.,[.,.]]]],[.,.]],.],.]
=> [6,4,3,2,1,5,7,8] => [6,4,3,2,1,5,7] => [2,4,5,6,7,3,1] => ? = 6
[[[[[.,[.,.]],[.,.]],[.,.]],.],.]
=> [6,4,2,1,3,5,7,8] => ? => ? => ? = 6
[[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7] => [3,4,5,6,7,2,1] => ? = 6
[[[[[.,[.,[.,.]]],[.,.]],.],.],.]
=> [5,3,2,1,4,6,7,8] => ? => ? => ? = 6
[[[[[.,[.,[.,[.,.]]]],.],.],.],.]
=> [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => ? = 6
[[[[[[.,[.,.]],[.,.]],.],.],.],.]
=> [4,2,1,3,5,6,7,8] => ? => ? => ? = 6
[[[[[[.,[.,[.,.]]],.],.],.],.],.]
=> [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => ? = 6
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.
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St000740: Permutations ⟶ ℤResult quality: 78% values known / values provided: 84%distinct values known / distinct values provided: 78%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => [1] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [2,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [2,1] => 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,2] => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [1,2,4,3] => 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [1,2,4,3] => 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [4,3,1,2] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [1,4,2,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [1,4,2,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [1,3,2,4] => 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [6,8,7,5,4,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [2,3,5,6,8,7,4,1] => [2,3,5,6,7,4,1] => [3,4,7,1,2,6,5] => ? = 4 + 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [2,3,5,6,7,8,4,1] => [2,3,5,6,7,4,1] => [3,4,7,1,2,6,5] => ? = 4 + 1
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [4,3,2,8,7,6,5,1] => [4,3,2,7,6,5,1] => [1,7,2,6,3,5,4] => ? = 3 + 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [2,3,4,8,7,6,5,1] => [2,3,4,7,6,5,1] => [7,6,1,2,3,5,4] => ? = 3 + 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [2,3,4,6,7,8,5,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [5,4,3,2,8,7,6,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [5,4,3,2,7,8,6,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [2,3,4,5,8,7,6,1] => ? => ? => ? = 4 + 1
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,7,8,6,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [6,5,4,3,2,8,7,1] => ? => ? => ? = 5 + 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 + 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [2,3,5,6,7,4,8,1] => ? => ? => ? = 4 + 1
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [4,3,2,7,6,5,8,1] => [4,3,2,7,6,5,1] => [1,7,2,6,3,5,4] => ? = 3 + 1
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [5,4,3,2,7,6,8,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [2,3,4,5,7,6,8,1] => ? => ? => ? = 4 + 1
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [2,3,6,5,4,7,8,1] => ? => ? => ? = 5 + 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [2,3,5,6,4,7,8,1] => ? => ? => ? = 5 + 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 1 + 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 1 + 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,7,6,5,8,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 1 + 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? => ? => ? = 1 + 1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [2,1,8,7,6,5,4,3] => [2,1,7,6,5,4,3] => [7,6,5,1,4,2,3] => ? = 2 + 1
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
[[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [4,3,2,1,7,8,6,5] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [4,3,2,1,7,6,8,5] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => [5,4,3,2,1,7,6] => [4,3,2,1,7,5,6] => ? = 5 + 1
[[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [5,4,3,2,1,7,8,6] => [5,4,3,2,1,7,6] => [4,3,2,1,7,5,6] => ? = 5 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 6 + 1
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [3,2,1,6,5,4,8,7] => ? => ? => ? = 6 + 1
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,7] => [3,2,1,6,4,5,7] => ? = 6 + 1
[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [5,4,3,2,1,6,8,7] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 6 + 1
[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [4,3,2,1,5,6,8,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 6 + 1
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 6 + 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [6,7,5,4,3,2,1,8] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 + 1
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [5,4,3,2,7,6,1,8] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [2,3,4,5,7,6,1,8] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 4 + 1
[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> [6,5,4,3,2,7,1,8] => ? => ? => ? = 5 + 1
[[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [2,1,7,6,5,4,3,8] => ? => ? => ? = 2 + 1
[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [4,3,2,1,7,6,5,8] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
[[[.,[.,[.,[.,[.,.]]]]],[.,.]],.]
=> [5,4,3,2,1,7,6,8] => [5,4,3,2,1,7,6] => [4,3,2,1,7,5,6] => ? = 5 + 1
[[[[.,[.,[.,.]]],[.,.]],[.,.]],.]
=> [3,2,1,5,4,7,6,8] => [3,2,1,5,4,7,6] => [2,1,4,7,3,5,6] => ? = 5 + 1
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Mp00017: Binary trees to 312-avoiding permutationPermutations
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St000727: Permutations ⟶ ℤResult quality: 78% values known / values provided: 84%distinct values known / distinct values provided: 78%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => ? = 0 + 1
[[.,.],.]
=> [1,2] => [1] => [1] => ? = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [2,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [2,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [2,1] => 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,2] => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [3,2,1] => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,1,3] => 3 = 2 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [1,2,4,3] => 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [1,2,4,3] => 3 = 2 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [4,3,1,2] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [1,4,2,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [1,4,2,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [3,2,1,4] => 4 = 3 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [1,3,2,4] => 4 = 3 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => 4 = 3 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 4 = 3 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => [1,4,3,2] => 2 = 1 + 1
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [2,4,3,1] => [4,1,3,2] => 2 = 1 + 1
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [6,8,7,5,4,3,2,1] => ? => ? => ? = 1 + 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 + 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [2,3,5,6,8,7,4,1] => [2,3,5,6,7,4,1] => [3,4,7,1,2,6,5] => ? = 4 + 1
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [2,3,5,6,7,8,4,1] => [2,3,5,6,7,4,1] => [3,4,7,1,2,6,5] => ? = 4 + 1
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [4,3,2,8,7,6,5,1] => [4,3,2,7,6,5,1] => [1,7,2,6,3,5,4] => ? = 3 + 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [2,3,4,8,7,6,5,1] => [2,3,4,7,6,5,1] => [7,6,1,2,3,5,4] => ? = 3 + 1
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [2,3,4,6,7,8,5,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [5,4,3,2,8,7,6,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [5,4,3,2,7,8,6,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [2,3,4,5,8,7,6,1] => ? => ? => ? = 4 + 1
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [2,3,4,5,7,8,6,1] => ? => ? => ? = 4 + 1
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [6,5,4,3,2,8,7,1] => ? => ? => ? = 5 + 1
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 + 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [2,3,5,6,7,4,8,1] => ? => ? => ? = 4 + 1
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [4,3,2,7,6,5,8,1] => [4,3,2,7,6,5,1] => [1,7,2,6,3,5,4] => ? = 3 + 1
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [5,4,3,2,7,6,8,1] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [2,3,4,5,7,6,8,1] => ? => ? => ? = 4 + 1
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [2,3,6,5,4,7,8,1] => ? => ? => ? = 5 + 1
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [2,3,5,6,4,7,8,1] => ? => ? => ? = 5 + 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 1 + 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 1 + 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [1,7,6,5,8,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 1 + 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => ? => ? => ? = 1 + 1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [2,1,8,7,6,5,4,3] => [2,1,7,6,5,4,3] => [7,6,5,1,4,2,3] => ? = 2 + 1
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [4,3,2,1,8,7,6,5] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
[[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [4,3,2,1,7,8,6,5] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [4,3,2,1,7,6,8,5] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [5,4,3,2,1,8,7,6] => [5,4,3,2,1,7,6] => [4,3,2,1,7,5,6] => ? = 5 + 1
[[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [5,4,3,2,1,7,8,6] => [5,4,3,2,1,7,6] => [4,3,2,1,7,5,6] => ? = 5 + 1
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 6 + 1
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [3,2,1,6,5,4,8,7] => ? => ? => ? = 6 + 1
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [4,3,2,1,6,5,8,7] => [4,3,2,1,6,5,7] => [3,2,1,6,4,5,7] => ? = 6 + 1
[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [5,4,3,2,1,6,8,7] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 6 + 1
[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [4,3,2,1,5,6,8,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 6 + 1
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 6 + 1
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0 + 1
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [6,7,5,4,3,2,1,8] => [6,7,5,4,3,2,1] => [1,7,6,5,4,3,2] => ? = 1 + 1
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [5,4,3,2,7,6,1,8] => [5,4,3,2,7,6,1] => [3,2,1,7,4,6,5] => ? = 4 + 1
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [2,3,4,5,7,6,1,8] => [2,3,4,5,7,6,1] => [7,1,2,3,4,6,5] => ? = 4 + 1
[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> [6,5,4,3,2,7,1,8] => ? => ? => ? = 5 + 1
[[[.,[.,.]],[.,[.,[.,[.,.]]]]],.]
=> [2,1,7,6,5,4,3,8] => ? => ? => ? = 2 + 1
[[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
=> [4,3,2,1,7,6,5,8] => [4,3,2,1,7,6,5] => [2,1,7,3,6,4,5] => ? = 4 + 1
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.
Mp00014: Binary trees to 132-avoiding permutationPermutations
Mp00064: Permutations reversePermutations
Mp00252: Permutations restrictionPermutations
St000316: Permutations ⟶ ℤResult quality: 67% values known / values provided: 84%distinct values known / distinct values provided: 67%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1] => 0
[[.,.],.]
=> [1,2] => [2,1] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,3,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,3] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2] => 2
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [2,1,3] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 2
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 3
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 3
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 3
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 3
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> [7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> [8,6,7,5,4,3,2,1] => [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6] => ? = 1
[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> [7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [1,2,3,4,5,8,7,6] => [1,2,3,4,5,7,6] => ? = 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [7,6,5,8,4,3,2,1] => [1,2,3,4,8,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> [7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[.,[[[.,.],.],[[[.,.],.],[.,.]]]]
=> [8,5,6,7,2,3,4,1] => [1,4,3,2,7,6,5,8] => [1,4,3,2,7,6,5] => ? = 4
[.,[[[.,.],.],[[[[.,.],.],.],.]]]
=> [5,6,7,8,2,3,4,1] => [1,4,3,2,8,7,6,5] => [1,4,3,2,7,6,5] => ? = 4
[.,[[.,[.,[.,.]]],[.,[.,[.,.]]]]]
=> [8,7,6,4,3,2,5,1] => [1,5,2,3,4,6,7,8] => [1,5,2,3,4,6,7] => ? = 3
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
=> [8,7,6,2,3,4,5,1] => [1,5,4,3,2,6,7,8] => ? => ? = 3
[.,[[[[.,.],.],.],[[[.,.],.],.]]]
=> [6,7,8,2,3,4,5,1] => [1,5,4,3,2,8,7,6] => ? => ? = 4
[.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> [8,7,5,4,3,2,6,1] => [1,6,2,3,4,5,7,8] => [1,6,2,3,4,5,7] => ? = 4
[.,[[.,[.,[.,[.,.]]]],[[.,.],.]]]
=> [7,8,5,4,3,2,6,1] => [1,6,2,3,4,5,8,7] => ? => ? = 4
[.,[[[[[.,.],.],.],.],[.,[.,.]]]]
=> [8,7,2,3,4,5,6,1] => [1,6,5,4,3,2,7,8] => ? => ? = 4
[.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [7,8,2,3,4,5,6,1] => [1,6,5,4,3,2,8,7] => ? => ? = 4
[.,[[.,[.,[.,[.,[.,.]]]]],[.,.]]]
=> [8,6,5,4,3,2,7,1] => [1,7,2,3,4,5,6,8] => [1,7,2,3,4,5,6] => ? = 5
[.,[[[[[[.,.],.],.],.],.],[.,.]]]
=> [8,2,3,4,5,6,7,1] => [1,7,6,5,4,3,2,8] => ? => ? = 5
[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]
=> [7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [1,8,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 1
[.,[[[[.,.],.],[[[.,.],.],.]],.]]
=> [5,6,7,2,3,4,8,1] => [1,8,4,3,2,7,6,5] => ? => ? = 4
[.,[[[.,[.,[.,.]]],[.,[.,.]]],.]]
=> [7,6,4,3,2,5,8,1] => [1,8,5,2,3,4,6,7] => ? => ? = 3
[.,[[[.,[.,[.,[.,.]]]],[.,.]],.]]
=> [7,5,4,3,2,6,8,1] => [1,8,6,2,3,4,5,7] => ? => ? = 4
[.,[[[[[[.,.],.],.],.],[.,.]],.]]
=> [7,2,3,4,5,6,8,1] => [1,8,6,5,4,3,2,7] => ? => ? = 4
[.,[[[[[.,.],.],[.,[.,.]]],.],.]]
=> [6,5,2,3,4,7,8,1] => [1,8,7,4,3,2,5,6] => [1,7,4,3,2,5,6] => ? = 5
[.,[[[[[.,.],.],[[.,.],.]],.],.]]
=> [5,6,2,3,4,7,8,1] => [1,8,7,4,3,2,6,5] => ? => ? = 5
[.,[[[[[[[.,.],.],.],.],.],.],.]]
=> [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => ? = 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,4,3,1,2] => [2,1,3,4,5,6,8,7] => [2,1,3,4,5,6,7] => ? = 1
[[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> [7,6,5,8,4,3,1,2] => [2,1,3,4,8,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [7,6,5,4,3,8,1,2] => [2,1,8,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7] => ? = 2
[[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> [8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => [5,1,2,3,4,6,7] => ? = 4
[[.,[.,[.,[.,.]]]],[.,[[.,.],.]]]
=> [7,8,6,4,3,2,1,5] => [5,1,2,3,4,6,8,7] => [5,1,2,3,4,6,7] => ? = 4
[[.,[.,[.,[.,.]]]],[[.,[.,.]],.]]
=> [7,6,8,4,3,2,1,5] => [5,1,2,3,4,8,6,7] => ? => ? = 4
[[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
=> [8,7,5,4,3,2,1,6] => [6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7] => ? = 5
[[.,[.,[.,[.,[.,.]]]]],[[.,.],.]]
=> [7,8,5,4,3,2,1,6] => [6,1,2,3,4,5,8,7] => [6,1,2,3,4,5,7] => ? = 5
[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> [8,6,5,4,3,2,1,7] => [7,1,2,3,4,5,6,8] => [7,1,2,3,4,5,6] => ? = 6
[[[.,[.,[.,.]]],[.,[.,.]]],[.,.]]
=> [8,6,5,3,2,1,4,7] => [7,4,1,2,3,5,6,8] => ? => ? = 6
[[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
=> [8,6,4,3,2,1,5,7] => [7,5,1,2,3,4,6,8] => [7,5,1,2,3,4,6] => ? = 6
[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]
=> [8,6,4,2,1,3,5,7] => [7,5,3,1,2,4,6,8] => [7,5,3,1,2,4,6] => ? = 6
[[[.,[.,[.,[.,[.,.]]]]],.],[.,.]]
=> [8,5,4,3,2,1,6,7] => [7,6,1,2,3,4,5,8] => ? => ? = 6
[[[[.,[.,[.,[.,.]]]],.],.],[.,.]]
=> [8,4,3,2,1,5,6,7] => [7,6,5,1,2,3,4,8] => [7,6,5,1,2,3,4] => ? = 6
[[[[[[.,[.,.]],.],.],.],.],[.,.]]
=> [8,2,1,3,4,5,6,7] => [7,6,5,4,3,1,2,8] => [7,6,5,4,3,1,2] => ? = 6
[[[[[[[.,.],.],.],.],.],.],[.,.]]
=> [8,1,2,3,4,5,6,7] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1] => ? = 6
[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> [7,6,5,4,3,2,1,8] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> [6,7,5,4,3,2,1,8] => [8,1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 1
[[.,[[.,[.,[.,[.,.]]]],[.,.]]],.]
=> [7,5,4,3,2,6,1,8] => [8,1,6,2,3,4,5,7] => ? => ? = 4
[[.,[[[[[.,.],.],.],.],[.,.]]],.]
=> [7,2,3,4,5,6,1,8] => [8,1,6,5,4,3,2,7] => ? => ? = 4
[[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> [6,5,4,3,2,7,1,8] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 5
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.
St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. 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.