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Your data matches 56 different statistics following compositions of up to 3 maps.
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Matching statistic: St000157
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [1,2] => [[1,2]]
=> 0
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,2,3] => [[1,2,3]]
=> 0
[2,1,3] => [1,2,3] => [[1,2,3]]
=> 0
[2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[3,2,1] => [1,3,2] => [[1,2],[3]]
=> 1
[1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,3,2,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,3,4,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,4,2,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,4,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,1,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,1,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,3,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[2,4,1,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[3,1,4,2] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[3,2,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[3,2,4,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[3,4,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[3,4,2,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,1,3,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[4,2,1,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,2,3,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[4,3,1,2] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[4,3,2,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [[1,2,3,5],[4]]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000288
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => => ? = 0
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,2,3] => [1,2,3] => 00 => 0
[2,1,3] => [1,2,3] => [1,2,3] => 00 => 0
[2,3,1] => [1,2,3] => [1,2,3] => 00 => 0
[3,1,2] => [1,3,2] => [1,3,2] => 01 => 1
[3,2,1] => [1,3,2] => [1,3,2] => 01 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 001 => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 001 => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 010 => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 011 => 2
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => 001 => 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 011 => 2
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 001 => 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 001 => 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 001 => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 0001 => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => 0001 => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [1,5,3,7,6,2,4,8] => ? => ? = 3
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [1,3,5,8,6,7,4,2] => ? => ? = 3
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [1,3,6,7,4,8,2,5] => ? => ? = 2
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [1,4,6,7,2,5,8,3] => ? => ? = 2
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [1,3,6,8,5,4,7,2] => ? => ? = 3
[8,6,4,5,3,7,2,1] => [1,8,2,6,7,3,4,5] => [1,3,6,7,8,4,5,2] => ? => ? = 2
[8,6,4,3,5,7,2,1] => [1,8,2,6,7,3,4,5] => [1,3,6,7,8,4,5,2] => ? => ? = 2
[6,5,7,4,3,8,2,1] => [1,6,8,2,5,3,7,4] => [1,4,6,8,5,2,7,3] => ? => ? = 3
[6,7,4,5,3,8,2,1] => [1,6,8,2,7,3,4,5] => [1,4,6,7,8,2,5,3] => ? => ? = 2
[6,5,4,7,3,8,2,1] => [1,6,8,2,5,3,4,7] => [1,4,6,7,5,2,8,3] => ? => ? = 3
[4,5,6,7,3,8,2,1] => [1,4,7,2,5,3,6,8] => [1,4,6,2,5,7,3,8] => ? => ? = 2
[7,5,4,3,6,8,2,1] => [1,7,2,5,6,8,3,4] => [1,3,7,8,4,5,2,6] => ? => ? = 2
[7,5,3,4,6,8,2,1] => [1,7,2,5,6,8,3,4] => [1,3,7,8,4,5,2,6] => ? => ? = 2
[4,3,5,6,7,8,2,1] => [1,4,6,8,2,3,5,7] => [1,5,6,2,7,3,8,4] => ? => ? = 3
[8,7,5,4,6,2,3,1] => [1,8,2,7,3,5,6,4] => [1,3,5,8,6,7,4,2] => ? => ? = 3
[7,8,5,6,2,3,4,1] => [1,7,4,6,3,5,2,8] => [1,7,5,3,6,4,2,8] => ? => ? = 4
[8,7,6,4,3,2,5,1] => [1,8,2,7,5,3,6,4] => [1,3,6,8,5,7,4,2] => ? => ? = 3
[8,6,7,4,3,2,5,1] => [1,8,2,6,3,7,5,4] => [1,3,5,8,7,4,6,2] => ? => ? = 3
[7,6,8,4,3,2,5,1] => [1,7,5,3,8,2,6,4] => [1,6,4,8,3,7,2,5] => ? => ? = 3
[8,7,6,3,4,2,5,1] => [1,8,2,7,5,4,3,6] => [1,3,7,6,5,8,4,2] => ? => ? = 4
[8,6,7,3,4,2,5,1] => [1,8,2,6,3,7,5,4] => [1,3,5,8,7,4,6,2] => ? => ? = 3
[8,7,6,4,2,3,5,1] => [1,8,2,7,5,3,6,4] => [1,3,6,8,5,7,4,2] => ? => ? = 3
[8,6,7,4,2,3,5,1] => [1,8,2,6,3,7,5,4] => [1,3,5,8,7,4,6,2] => ? => ? = 3
[6,7,8,4,2,3,5,1] => [1,6,3,8,2,7,5,4] => [1,5,3,8,7,2,6,4] => ? => ? = 4
[8,7,6,3,2,4,5,1] => [1,8,2,7,5,3,6,4] => [1,3,6,8,5,7,4,2] => ? => ? = 3
[8,7,6,2,3,4,5,1] => [1,8,2,7,5,3,6,4] => [1,3,6,8,5,7,4,2] => ? => ? = 3
[7,6,8,2,3,4,5,1] => [1,7,5,3,8,2,6,4] => [1,6,4,8,3,7,2,5] => ? => ? = 3
[7,8,5,4,3,2,6,1] => [1,7,6,2,8,3,5,4] => [1,4,6,8,7,3,2,5] => ? => ? = 3
[7,8,5,3,4,2,6,1] => [1,7,6,2,8,3,5,4] => [1,4,6,8,7,3,2,5] => ? => ? = 3
[8,7,5,3,2,4,6,1] => [1,8,2,7,6,4,3,5] => [1,3,7,6,8,5,4,2] => ? => ? = 4
[8,6,4,3,2,5,7,1] => [1,8,2,6,5,3,4,7] => [1,3,6,7,5,4,8,2] => ? => ? = 3
[8,6,4,2,3,5,7,1] => [1,8,2,6,5,3,4,7] => [1,3,6,7,5,4,8,2] => ? => ? = 3
[5,6,7,3,4,2,8,1] => [1,5,4,3,7,8,2,6] => [1,7,4,3,2,8,5,6] => ? => ? = 4
[7,4,5,3,6,2,8,1] => [1,7,8,2,4,3,5,6] => [1,4,6,5,7,8,2,3] => ? => ? = 2
[6,7,5,4,2,3,8,1] => [1,6,3,5,2,7,8,4] => [1,5,3,8,4,2,6,7] => ? => ? = 3
[7,5,4,6,2,3,8,1] => [1,7,8,2,5,3,4,6] => [1,4,6,7,5,8,2,3] => ? => ? = 2
[7,5,4,3,2,6,8,1] => [1,7,8,2,5,3,4,6] => [1,4,6,7,5,8,2,3] => ? => ? = 2
[7,5,4,2,3,6,8,1] => [1,7,8,2,5,3,4,6] => [1,4,6,7,5,8,2,3] => ? => ? = 2
[7,4,5,2,3,6,8,1] => [1,7,8,2,4,3,5,6] => [1,4,6,5,7,8,2,3] => ? => ? = 2
[7,4,3,2,5,6,8,1] => [1,7,8,2,4,3,5,6] => [1,4,6,5,7,8,2,3] => ? => ? = 2
[7,4,2,3,5,6,8,1] => [1,7,8,2,4,3,5,6] => [1,4,6,5,7,8,2,3] => ? => ? = 2
[6,5,7,8,4,3,1,2] => [1,6,3,7,2,5,4,8] => [1,5,3,7,6,2,4,8] => ? => ? = 3
[8,7,5,4,6,3,1,2] => [1,8,2,7,3,5,6,4] => [1,3,5,8,6,7,4,2] => ? => ? = 3
[5,6,4,7,8,3,1,2] => [1,5,8,2,6,3,4,7] => [1,4,6,7,2,5,8,3] => ? => ? = 2
[6,5,7,4,3,8,1,2] => [1,6,8,2,5,3,7,4] => [1,4,6,8,5,2,7,3] => ? => ? = 3
[6,7,4,5,3,8,1,2] => [1,6,8,2,7,3,4,5] => [1,4,6,7,8,2,5,3] => ? => ? = 2
[6,5,4,7,3,8,1,2] => [1,6,8,2,5,3,4,7] => [1,4,6,7,5,2,8,3] => ? => ? = 3
[4,5,6,7,3,8,1,2] => [1,4,7,2,5,3,6,8] => [1,4,6,2,5,7,3,8] => ? => ? = 2
[7,5,4,3,6,8,1,2] => [1,7,2,5,6,8,3,4] => [1,3,7,8,4,5,2,6] => ? => ? = 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000052
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 3
[8,5,6,7,4,3,2,1] => [1,8,2,5,4,7,3,6] => [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
[7,5,6,8,4,3,2,1] => [1,7,2,5,4,8,3,6] => [.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3
[5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,5,6,4,7,3,2,1] => [1,8,2,5,7,3,6,4] => [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2
[6,7,4,5,8,3,2,1] => [1,6,3,4,5,8,2,7] => [.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[7,4,5,6,8,3,2,1] => [1,7,2,4,6,3,5,8] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
[6,4,5,7,8,3,2,1] => [1,6,3,5,8,2,4,7] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 4
[5,4,6,7,8,3,2,1] => [1,5,8,2,4,7,3,6] => [.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[6,7,5,8,3,4,2,1] => [1,6,4,8,2,7,3,5] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3
[6,5,7,8,3,4,2,1] => [1,6,4,8,2,5,3,7] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[8,7,6,4,3,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[7,8,6,4,3,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[6,7,8,4,3,5,2,1] => [1,6,5,3,8,2,7,4] => [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 4
[8,7,6,3,4,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[7,8,6,3,4,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3
[7,6,8,3,4,5,2,1] => [1,7,2,6,5,4,3,8] => [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000167
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000167: Ordered trees ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [[]]
=> 1 = 0 + 1
[1,2] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [[[]]]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [[[],[]]]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [[[[[[]],[]],[]],[]]]
=> ? = 3 + 1
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [[[[[[]],[]]],[[]]]]
=> ? = 2 + 1
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [[[[[[]]],[[]]],[]]]
=> ? = 2 + 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [[[[[[]]],[]],[[]]]]
=> ? = 2 + 1
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [[[[[],[[]]],[]],[]]]
=> ? = 3 + 1
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [[[[[],[[]]]],[[]]]]
=> ? = 2 + 1
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [[[[[],[]],[[]]],[]]]
=> ? = 3 + 1
[8,5,6,7,4,3,2,1] => [1,8,2,5,4,7,3,6] => [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [[[[],[],[[],[]]],[]]]
=> ? = 4 + 1
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [[[[[],[]],[]],[[]]]]
=> ? = 3 + 1
[7,5,6,8,4,3,2,1] => [1,7,2,5,4,8,3,6] => [.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [[[[],[],[[]]],[[]]]]
=> ? = 3 + 1
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [[[],[[],[]],[[[]]]]]
=> ? = 3 + 1
[5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [[[[]],[],[[[]],[]]]]
=> ? = 3 + 1
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [[[[[[]],[]],[]],[]]]
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [[[[[[]],[]]],[[]]]]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [[[[[[]]],[[]]],[]]]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [[[[[[]]],[]],[[]]]]
=> ? = 2 + 1
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [[[[[],[[]]],[]],[]]]
=> ? = 3 + 1
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [[[[[[[]]]],[]],[]]]
=> ? = 2 + 1
[8,5,6,4,7,3,2,1] => [1,8,2,5,7,3,6,4] => [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [[[[[]],[[],[]]],[]]]
=> ? = 3 + 1
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [[[[[[]]],[[]]],[]]]
=> ? = 2 + 1
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [[[[[[]]],[]],[[]]]]
=> ? = 2 + 1
[6,7,4,5,8,3,2,1] => [1,6,3,4,5,8,2,7] => [.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [[[],[[[]]],[[],[]]]]
=> ? = 3 + 1
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [[[[[]],[[]]],[[]]]]
=> ? = 2 + 1
[7,4,5,6,8,3,2,1] => [1,7,2,4,6,3,5,8] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]]
=> [[[[],[[],[]]],[[]]]]
=> ? = 3 + 1
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [[[[[]]],[[[]],[]]]]
=> ? = 2 + 1
[6,4,5,7,8,3,2,1] => [1,6,3,5,8,2,4,7] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [[[],[[],[]],[[],[]]]]
=> ? = 4 + 1
[5,4,6,7,8,3,2,1] => [1,5,8,2,4,7,3,6] => [.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [[[[],[]],[[],[],[]]]]
=> ? = 4 + 1
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [[[[]],[[[]],[[]]]]]
=> ? = 2 + 1
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [[[[[[]],[]],[]],[]]]
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [[[[[[]],[]]],[[]]]]
=> ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [[[[[],[[]]],[]],[]]]
=> ? = 3 + 1
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [[[[[],[[]]]],[[]]]]
=> ? = 2 + 1
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [[[[[]],[[[]]]],[]]]
=> ? = 2 + 1
[6,7,5,8,3,4,2,1] => [1,6,4,8,2,7,3,5] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [[[[]],[[]],[[],[]]]]
=> ? = 3 + 1
[6,5,7,8,3,4,2,1] => [1,6,4,8,2,5,3,7] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [[[[]],[[]],[[],[]]]]
=> ? = 3 + 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [[[],[[]],[[],[[]]]]]
=> ? = 3 + 1
[8,7,6,4,3,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [[[[[],[],[]],[]],[]]]
=> ? = 4 + 1
[7,8,6,4,3,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [[[[[],[],[]]],[[]]]]
=> ? = 3 + 1
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [[[[[]],[],[[]]],[]]]
=> ? = 3 + 1
[6,7,8,4,3,5,2,1] => [1,6,5,3,8,2,7,4] => [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [[[],[[]],[],[[],[]]]]
=> ? = 4 + 1
[8,7,6,3,4,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [[[[[],[],[]],[]],[]]]
=> ? = 4 + 1
[7,8,6,3,4,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [[[[[],[],[]]],[[]]]]
=> ? = 3 + 1
[7,6,8,3,4,5,2,1] => [1,7,2,6,5,4,3,8] => [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [[[[],[],[],[]],[[]]]]
=> ? = 4 + 1
[6,7,8,3,4,5,2,1] => [1,6,5,4,3,8,2,7] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [[[],[],[],[],[[],[]]]]
=> ? = 5 + 1
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [[[[[],[[]]],[]],[]]]
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [[[[[],[[]]]],[[]]]]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [[[[[[[]]]],[]],[]]]
=> ? = 2 + 1
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [[[[[],[[]]],[]],[]]]
=> ? = 3 + 1
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [[[[[],[[]]]],[[]]]]
=> ? = 2 + 1
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [[[[[[[]]]],[]],[]]]
=> ? = 2 + 1
Description
The number of leaves of an ordered tree.
This is the number of nodes which do not have any children.
Matching statistic: St000024
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[8,5,6,7,4,3,2,1] => [1,8,2,5,4,7,3,6] => [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 4
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 3
[7,5,6,8,4,3,2,1] => [1,7,2,5,4,8,3,6] => [.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 3
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 3
[5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 3
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[8,5,6,4,7,3,2,1] => [1,8,2,5,7,3,6,4] => [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 3
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2
[6,7,4,5,8,3,2,1] => [1,6,3,4,5,8,2,7] => [.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 3
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2
[7,4,5,6,8,3,2,1] => [1,7,2,4,6,3,5,8] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2
[6,4,5,7,8,3,2,1] => [1,6,3,5,8,2,4,7] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 4
[5,4,6,7,8,3,2,1] => [1,5,8,2,4,7,3,6] => [.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2
[6,7,5,8,3,4,2,1] => [1,6,4,8,2,7,3,5] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3
[6,5,7,8,3,4,2,1] => [1,6,4,8,2,5,3,7] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3
[8,7,6,4,3,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4
[7,8,6,4,3,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 3
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[6,7,8,4,3,5,2,1] => [1,6,5,3,8,2,7,4] => [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 4
[8,7,6,3,4,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4
[7,8,6,3,4,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 3
[7,6,8,3,4,5,2,1] => [1,7,2,6,5,4,3,8] => [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4
[6,7,8,3,4,5,2,1] => [1,6,5,4,3,8,2,7] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000053
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 3
[8,5,6,7,4,3,2,1] => [1,8,2,5,4,7,3,6] => [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3
[7,5,6,8,4,3,2,1] => [1,7,2,5,4,8,3,6] => [.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3
[5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,5,6,4,7,3,2,1] => [1,8,2,5,7,3,6,4] => [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2
[6,7,4,5,8,3,2,1] => [1,6,3,4,5,8,2,7] => [.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[7,4,5,6,8,3,2,1] => [1,7,2,4,6,3,5,8] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 2
[6,4,5,7,8,3,2,1] => [1,6,3,5,8,2,4,7] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 4
[5,4,6,7,8,3,2,1] => [1,5,8,2,4,7,3,6] => [.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[6,7,5,8,3,4,2,1] => [1,6,4,8,2,7,3,5] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3
[6,5,7,8,3,4,2,1] => [1,6,4,8,2,5,3,7] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[8,7,6,4,3,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[7,8,6,4,3,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3
[6,7,8,4,3,5,2,1] => [1,6,5,3,8,2,7,4] => [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 4
[8,7,6,3,4,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4
[7,8,6,3,4,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3
[7,6,8,3,4,5,2,1] => [1,7,2,6,5,4,3,8] => [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4
[6,7,8,3,4,5,2,1] => [1,6,5,4,3,8,2,7] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2
Description
The number of valleys of the Dyck path.
Matching statistic: St001007
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 1
[8,5,6,7,4,3,2,1] => [1,8,2,5,4,7,3,6] => [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 4 + 1
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 3 + 1
[7,5,6,8,4,3,2,1] => [1,7,2,5,4,8,3,6] => [.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> ? = 3 + 1
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[8,5,6,4,7,3,2,1] => [1,8,2,5,7,3,6,4] => [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[6,7,4,5,8,3,2,1] => [1,6,3,4,5,8,2,7] => [.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> ? = 3 + 1
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[7,4,5,6,8,3,2,1] => [1,7,2,4,6,3,5,8] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3 + 1
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 2 + 1
[6,4,5,7,8,3,2,1] => [1,6,3,5,8,2,4,7] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> ? = 4 + 1
[5,4,6,7,8,3,2,1] => [1,5,8,2,4,7,3,6] => [.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,0,1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 4 + 1
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 2 + 1
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[6,7,5,8,3,4,2,1] => [1,6,4,8,2,7,3,5] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3 + 1
[6,5,7,8,3,4,2,1] => [1,6,4,8,2,5,3,7] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> ? = 3 + 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> ? = 3 + 1
[8,7,6,4,3,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[7,8,6,4,3,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 3 + 1
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 + 1
[6,7,8,4,3,5,2,1] => [1,6,5,3,8,2,7,4] => [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 4 + 1
[8,7,6,3,4,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[7,8,6,3,4,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 3 + 1
[7,6,8,3,4,5,2,1] => [1,7,2,6,5,4,3,8] => [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 4 + 1
[6,7,8,3,4,5,2,1] => [1,6,5,4,3,8,2,7] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 5 + 1
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001068
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 1 = 0 + 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 1
[8,5,6,7,4,3,2,1] => [1,8,2,5,4,7,3,6] => [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4 + 1
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[7,5,6,8,4,3,2,1] => [1,7,2,5,4,8,3,6] => [.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 + 1
[5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,5,6,4,7,3,2,1] => [1,8,2,5,7,3,6,4] => [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 + 1
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[6,7,4,5,8,3,2,1] => [1,6,3,4,5,8,2,7] => [.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[7,4,5,6,8,3,2,1] => [1,7,2,4,6,3,5,8] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[6,4,5,7,8,3,2,1] => [1,6,3,5,8,2,4,7] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 4 + 1
[5,4,6,7,8,3,2,1] => [1,5,8,2,4,7,3,6] => [.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 1
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
[6,7,5,8,3,4,2,1] => [1,6,4,8,2,7,3,5] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[6,5,7,8,3,4,2,1] => [1,6,4,8,2,5,3,7] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
[8,7,6,4,3,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 1
[7,8,6,4,3,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 1
[6,7,8,4,3,5,2,1] => [1,6,5,3,8,2,7,4] => [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 4 + 1
[8,7,6,3,4,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 1
[7,8,6,3,4,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[7,6,8,3,4,5,2,1] => [1,7,2,6,5,4,3,8] => [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[6,7,8,3,4,5,2,1] => [1,6,5,4,3,8,2,7] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
Description
Number of torsionless simple modules in the corresponding Nakayama algebra.
Matching statistic: St001499
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [.,.]
=> [1,0]
=> ? = 0 + 1
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [.,[[.,[[.,[[.,.],.]],[.,.]]],.]]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 1
[8,5,6,7,4,3,2,1] => [1,8,2,5,4,7,3,6] => [.,[[.,[[[.,.],.],[[.,.],.]]],.]]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 4 + 1
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
=> [1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[7,5,6,8,4,3,2,1] => [1,7,2,5,4,8,3,6] => [.,[[.,[[[.,.],.],[.,.]]],[.,.]]]
=> [1,1,1,0,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[6,5,7,8,4,3,2,1] => [1,6,3,7,2,5,4,8] => [.,[[[.,.],[[.,.],.]],[.,[.,.]]]]
=> [1,1,0,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 + 1
[5,6,7,8,4,3,2,1] => [1,5,4,8,2,6,3,7] => [.,[[[.,[.,.]],.],[[.,[.,.]],.]]]
=> [1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,5,4,6,3,2,1] => [1,8,2,7,3,5,6,4] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,5,6,4,7,3,2,1] => [1,8,2,5,7,3,6,4] => [.,[[.,[[.,[.,.]],[[.,.],.]]],.]]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 + 1
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [.,[[.,[[.,[.,[.,.]]],[.,.]]],.]]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [.,[[.,[[.,[.,[.,.]]],.]],[.,.]]]
=> [1,1,1,1,1,0,0,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[6,7,4,5,8,3,2,1] => [1,6,3,4,5,8,2,7] => [.,[[[.,.],[.,[.,.]]],[[.,.],.]]]
=> [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[7,5,4,6,8,3,2,1] => [1,7,2,5,8,3,4,6] => [.,[[.,[[.,[.,.]],[.,.]]],[.,.]]]
=> [1,1,1,1,0,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[7,4,5,6,8,3,2,1] => [1,7,2,4,6,3,5,8] => [.,[[.,[[.,.],[[.,.],.]]],[.,.]]]
=> [1,1,1,0,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[5,6,4,7,8,3,2,1] => [1,5,8,2,6,3,4,7] => [.,[[.,[.,[.,.]]],[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 2 + 1
[6,4,5,7,8,3,2,1] => [1,6,3,5,8,2,4,7] => [.,[[[.,.],[[.,.],.]],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 4 + 1
[5,4,6,7,8,3,2,1] => [1,5,8,2,4,7,3,6] => [.,[[.,[[.,.],.]],[[[.,.],.],.]]]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
[4,5,6,7,8,3,2,1] => [1,4,7,2,5,8,3,6] => [.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 2 + 1
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [.,[[.,[.,[[.,[.,.]],.]]],[.,.]]]
=> [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [.,[[.,[[.,[.,.]],[.,[.,.]]]],.]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
[6,7,5,8,3,4,2,1] => [1,6,4,8,2,7,3,5] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[6,5,7,8,3,4,2,1] => [1,6,4,8,2,5,3,7] => [.,[[[.,[.,.]],[.,.]],[[.,.],.]]]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [.,[[[.,.],[.,.]],[[.,.],[.,.]]]]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
[8,7,6,4,3,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 1
[7,8,6,4,3,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[8,6,7,4,3,5,2,1] => [1,8,2,6,5,3,7,4] => [.,[[.,[[[.,[.,.]],.],[.,.]]],.]]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 + 1
[6,7,8,4,3,5,2,1] => [1,6,5,3,8,2,7,4] => [.,[[[[.,.],[.,.]],.],[[.,.],.]]]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 4 + 1
[8,7,6,3,4,5,2,1] => [1,8,2,7,3,6,5,4] => [.,[[.,[[.,[[[.,.],.],.]],.]],.]]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 1
[7,8,6,3,4,5,2,1] => [1,7,2,8,3,6,5,4] => [.,[[.,[.,[[[.,.],.],.]]],[.,.]]]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[7,6,8,3,4,5,2,1] => [1,7,2,6,5,4,3,8] => [.,[[.,[[[[.,.],.],.],.]],[.,.]]]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[6,7,8,3,4,5,2,1] => [1,6,5,4,3,8,2,7] => [.,[[[[[.,.],.],.],.],[[.,.],.]]]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [.,[[.,[[.,[[.,.],[.,.]]],.]],.]]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [.,[[.,[.,[[.,.],[.,.]]]],[.,.]]]
=> [1,1,1,1,0,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2 + 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra.
We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000703
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[3,1,5,2,6,4,7] => [1,3,5,6,4,2,7] => [1,6,2,5,3,4,7] => [1,5,4,6,3,2,7] => ? = 2
[3,1,5,2,6,7,4] => [1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [1,6,4,5,7,3,2] => ? = 2
[3,1,5,2,7,6,4] => [1,3,5,7,4,2,6] => [1,6,2,5,3,7,4] => [1,5,6,7,3,2,4] => ? = 3
[3,1,5,4,6,2,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,5,4,6,2,3,7] => ? = 2
[3,1,5,4,6,7,2] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => [1,6,4,5,7,2,3] => ? = 2
[3,1,5,4,7,2,6] => [1,3,5,7,6,2,4] => [1,6,2,7,3,5,4] => [1,6,5,7,4,2,3] => ? = 3
[3,1,5,6,7,2,4] => [1,3,5,7,4,6,2] => [1,7,2,5,3,6,4] => [1,5,6,7,3,4,2] => ? = 3
[3,1,5,7,6,2,4] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,5,4,6,2,3,7] => ? = 2
[3,1,5,7,6,4,2] => [1,3,5,6,4,7,2] => [1,7,2,5,3,4,6] => [1,5,4,6,3,7,2] => ? = 2
[3,1,6,2,4,5,7] => [1,3,6,5,4,2,7] => [1,6,2,5,4,3,7] => [1,5,6,3,4,2,7] => ? = 3
[3,1,6,2,4,7,5] => [1,3,6,7,5,4,2] => [1,7,2,6,5,3,4] => [1,6,4,7,3,5,2] => ? = 3
[3,1,6,2,5,7,4] => [1,3,6,7,4,2,5] => [1,6,2,5,7,3,4] => [1,5,4,7,6,2,3] => ? = 2
[3,1,6,2,7,5,4] => [1,3,6,5,7,4,2] => [1,7,2,6,4,3,5] => [1,6,5,3,7,4,2] => ? = 3
[3,1,6,4,2,5,7] => [1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => [1,5,6,3,2,4,7] => ? = 3
[3,1,6,4,2,7,5] => [1,3,6,7,5,2,4] => [1,6,2,7,5,3,4] => [1,6,4,7,3,2,5] => ? = 3
[3,1,6,4,5,7,2] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => [1,5,4,7,2,6,3] => ? = 2
[3,1,6,4,7,5,2] => [1,3,6,5,7,2,4] => [1,6,2,7,4,3,5] => [1,6,5,3,7,2,4] => ? = 3
[3,1,6,5,4,7,2] => [1,3,6,7,2,4,5] => [1,5,2,6,7,3,4] => [1,5,4,7,2,6,3] => ? = 2
[3,1,6,7,2,5,4] => [1,3,6,5,2,4,7] => [1,5,2,6,4,3,7] => [1,5,6,3,2,4,7] => ? = 3
[3,1,6,7,4,5,2] => [1,3,6,5,4,7,2] => [1,7,2,5,4,3,6] => [1,5,6,3,4,7,2] => ? = 3
[3,1,7,2,4,5,6] => [1,3,7,6,5,4,2] => [1,7,2,6,5,4,3] => [1,6,7,3,4,5,2] => ? = 4
[3,1,7,2,4,6,5] => [1,3,7,5,4,2,6] => [1,6,2,5,4,7,3] => [1,5,7,6,4,2,3] => ? = 3
[3,1,7,2,5,4,6] => [1,3,7,6,4,2,5] => [1,6,2,5,7,4,3] => [1,5,7,3,6,2,4] => ? = 3
[3,1,7,2,5,6,4] => [1,3,7,4,2,5,6] => [1,5,2,4,6,7,3] => [1,4,7,5,2,6,3] => ? = 2
[3,1,7,2,6,4,5] => [1,3,7,5,6,4,2] => [1,7,2,6,4,5,3] => [1,6,7,5,3,4,2] => ? = 3
[3,1,7,2,6,5,4] => [1,3,7,4,2,5,6] => [1,5,2,4,6,7,3] => [1,4,7,5,2,6,3] => ? = 2
[3,1,7,4,2,5,6] => [1,3,7,6,5,2,4] => [1,6,2,7,5,4,3] => [1,6,7,3,4,2,5] => ? = 4
[3,1,7,4,2,6,5] => [1,3,7,5,2,4,6] => [1,5,2,6,4,7,3] => [1,5,7,6,2,4,3] => ? = 3
[3,1,7,4,5,2,6] => [1,3,7,6,2,4,5] => [1,5,2,6,7,4,3] => [1,5,7,3,2,6,4] => ? = 3
[3,1,7,4,5,6,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,4,7,2,5,6,3] => ? = 2
[3,1,7,4,6,2,5] => [1,3,7,5,6,2,4] => [1,6,2,7,4,5,3] => [1,6,7,5,3,2,4] => ? = 3
[3,1,7,4,6,5,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,4,7,2,5,6,3] => ? = 2
[3,1,7,5,2,4,6] => [1,3,7,6,4,5,2] => [1,7,2,5,6,4,3] => [1,5,7,3,6,4,2] => ? = 3
[3,1,7,5,2,6,4] => [1,3,7,4,5,2,6] => [1,6,2,4,5,7,3] => [1,4,7,5,6,2,3] => ? = 2
[3,1,7,5,4,2,6] => [1,3,7,6,2,4,5] => [1,5,2,6,7,4,3] => [1,5,7,3,2,6,4] => ? = 3
[3,1,7,5,4,6,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,4,7,2,5,6,3] => ? = 2
[3,1,7,5,6,2,4] => [1,3,7,4,5,6,2] => [1,7,2,4,5,6,3] => [1,4,7,5,6,3,2] => ? = 2
[3,1,7,5,6,4,2] => [1,3,7,2,4,5,6] => [1,4,2,5,6,7,3] => [1,4,7,2,5,6,3] => ? = 2
[3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,5,2,6,4,7,3] => [1,5,7,6,2,4,3] => ? = 3
[3,1,7,6,2,5,4] => [1,3,7,4,6,5,2] => [1,7,2,4,6,5,3] => [1,4,7,6,3,5,2] => ? = 3
[3,1,7,6,4,5,2] => [1,3,7,2,4,6,5] => [1,4,2,5,7,6,3] => [1,4,7,2,5,3,6] => ? = 3
[3,1,7,6,5,2,4] => [1,3,7,4,6,2,5] => [1,6,2,4,7,5,3] => [1,4,7,6,3,2,5] => ? = 3
[3,1,7,6,5,4,2] => [1,3,7,2,4,6,5] => [1,4,2,5,7,6,3] => [1,4,7,2,5,3,6] => ? = 3
[3,2,5,1,6,4,7] => [1,3,5,6,4,2,7] => [1,6,2,5,3,4,7] => [1,5,4,6,3,2,7] => ? = 2
[3,2,5,1,6,7,4] => [1,3,5,6,7,4,2] => [1,7,2,6,3,4,5] => [1,6,4,5,7,3,2] => ? = 2
[3,2,5,1,7,6,4] => [1,3,5,7,4,2,6] => [1,6,2,5,3,7,4] => [1,5,6,7,3,2,4] => ? = 3
[3,2,5,4,6,1,7] => [1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,5,4,6,2,3,7] => ? = 2
[3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => [1,6,4,5,7,2,3] => ? = 2
[3,2,5,4,7,1,6] => [1,3,5,7,6,2,4] => [1,6,2,7,3,5,4] => [1,6,5,7,4,2,3] => ? = 3
[3,2,5,6,7,1,4] => [1,3,5,7,4,6,2] => [1,7,2,5,3,6,4] => [1,5,6,7,3,4,2] => ? = 3
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
The following 46 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000354The number of recoils of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001427The number of descents of a signed permutation. St001896The number of right descents of a signed permutations. St001960The number of descents of a permutation minus one if its first entry is not one. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001624The breadth of a lattice. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function.
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