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Your data matches 125 different statistics following compositions of up to 3 maps.
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Matching statistic: St000292
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 010 => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 0001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 1
Description
The number of ascents of a binary word.
Matching statistic: St000389
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 010 => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 0001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 1
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000390
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 010 => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 001 => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 0001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 00 => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 01 => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 000 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 000 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 000 => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 001 => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 010 => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 010 => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 010 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 0010 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 0010 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 0010 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 0010 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 0010 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 0100 => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,8,10,11,9] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9,11] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,6,7,8,9,1,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => 0000000000 => ? = 0
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St001214
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,1,1]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,1,1]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [2,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [2,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [2,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [2,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [2,1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,2,3,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,2,5,4,3,7,1,8] => [1,6,7,2,3,5,4,8] => ?
=> ? = 2
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,3,2,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> [6,3,5,4,2,7,1,8] => [1,6,7,2,3,5,4,8] => ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,8,10,11,9] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,11,10,9] => [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => [1,3,4,5,6,7,8,9,10,11,2] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,8,9,11,10,1] => [1,2,3,4,5,6,7,8,9,11,10] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [11,2,3,4,5,6,7,8,9,10,1] => [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9,11] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,9,10,1,11] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [11,3,4,5,6,7,8,9,10,2,1] => [1,11,2,3,4,5,6,7,8,9,10] => [2,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6,9,8,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,2,4,1,7,6,8,5,11,10,12,9] => [1,3,4,2,5,7,8,6,9,11,12,10] => [2,2,2,1,1,1,1,1,1]
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,6,7,8,9,1,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St000157
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,1,0,0]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [[1,2],[3]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,6,5,7,1,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 1
[1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,5,1,8,7,6] => [1,3,4,5,2,6,8,7] => ?
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,2,3,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> [5,2,3,4,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> [5,2,4,3,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,3,2,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [5,3,2,4,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [5,3,4,2,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [8,5,3,4,7,6,2,1] => [1,8,2,5,7,3,4,6] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [8,5,4,3,7,6,2,1] => [1,8,2,5,7,3,4,6] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,9,7,8,1] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,6,9,8,7,1] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,5,9,6,7,8,1] => [1,2,3,4,5,9,6,7,8] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,4,5,9,6,8,7,1] => [1,2,3,4,5,9,6,7,8] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,4,5,9,7,6,8,1] => [1,2,3,4,5,9,6,7,8] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,9,7,8,6,1] => [1,2,3,4,5,9,6,7,8] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,4,9,5,6,7,8,1] => [1,2,3,4,9,5,6,7,8] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,4,9,6,5,7,8,1] => [1,2,3,4,9,5,6,7,8] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,5,1] => [1,2,3,4,9,5,6,7,8] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,3,9,4,5,6,7,8,1] => [1,2,3,9,4,5,6,7,8] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,3,9,5,4,6,7,8,1] => [1,2,3,9,4,5,6,7,8] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,3,9,5,6,7,8,4,1] => [1,2,3,9,4,5,6,7,8] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,9,3,4,5,6,7,8,1] => [1,2,9,3,4,5,6,7,8] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [2,9,4,3,5,6,7,8,1] => [1,2,9,3,4,5,6,7,8] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [2,9,4,5,6,7,8,3,1] => [1,2,9,3,4,5,6,7,8] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,2,9,4,5,6,7,8,1] => [1,3,9,2,4,5,6,7,8] => [[1,2,3,6,7,8,9],[4,5]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [7,2,3,4,5,6,9,8,1] => [1,7,9,2,3,4,5,6,8] => [[1,2,3,6,7,8,9],[4,5]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,2,3,4,5,6,7,9,1] => [1,8,9,2,3,4,5,6,7] => [[1,2,3,6,7,8,9],[4,5]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [9,2,4,5,6,8,7,3,1] => [1,9,2,3,4,5,6,8,7] => [[1,2,4,5,6,7,8],[3],[9]]
=> ? = 2
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [9,2,7,6,5,4,3,8,1] => [1,9,2,3,7,4,6,5,8] => [[1,2,4,5,7,9],[3],[6],[8]]
=> ? = 3
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [9,3,4,5,6,8,7,2,1] => [1,9,2,3,4,5,6,8,7] => [[1,2,4,5,6,7,8],[3],[9]]
=> ? = 2
[1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
=> [9,4,3,5,6,7,8,2,1] => [1,9,2,4,5,6,7,8,3] => ?
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,7,8,10,11,9] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,5,9,7,8,6] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,5,9,8,7,6] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,11,2] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [1,2,3,4,5,6,7,8,9,10,11] => [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,6,7,8,11,10,9] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,4,9,6,7,8,5] => [1,2,3,4,5,9,6,7,8] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,11,1] => [1,3,4,5,6,7,8,9,10,11,2] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,2,3,9,5,6,7,8,4] => [1,2,3,4,9,5,6,7,8] => [[1,2,3,4,5,7,8,9],[6]]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,9,8,7,6,5] => [1,2,3,4,5,9,6,8,7] => [[1,2,3,4,5,6,8],[7],[9]]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,9,4,5,6,7,8,3] => [1,2,3,9,4,5,6,7,8] => [[1,2,3,4,6,7,8,9],[5]]
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,9,3,4,5,6,7,8,2] => [1,2,9,3,4,5,6,7,8] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,9,4,5,6,7,8,3,2] => [1,2,9,3,4,5,6,7,8] => [[1,2,3,5,6,7,8,9],[4]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [9,2,3,4,5,6,7,8,10,1] => [1,9,10,2,3,4,5,6,7,8] => [[1,2,3,6,7,8,9,10],[4,5]]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,2,3,5,4,9,8,7,6] => [1,2,3,4,5,6,9,7,8] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 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: St000919
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0
[1,1,0,0]
=> [2,1] => [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,3,6,5,7,8,2] => [1,2,4,6,7,8,3,5] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,6,5,7,1,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,6,5,7,8,1] => [1,2,4,6,7,8,3,5] => ?
=> ? = 1
[1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,5,1,8,7,6] => [1,3,4,5,2,6,8,7] => ?
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,2,3,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> [5,2,3,4,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> [5,2,4,3,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,3,2,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [5,3,2,4,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [5,3,4,2,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [7,4,3,6,5,2,8,1] => [1,7,8,2,4,6,3,5] => ?
=> ? = 2
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [8,5,3,4,7,6,2,1] => [1,8,2,5,7,3,4,6] => ?
=> ? = 2
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [8,5,4,3,7,6,2,1] => [1,8,2,5,7,3,4,6] => ?
=> ? = 2
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,9,8,1] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,5,6,8,7,9,1] => [1,2,3,4,5,6,8,9,7] => [.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,9,7,8,1] => [1,2,3,4,5,6,9,7,8] => [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,6,9,8,7,1] => [1,2,3,4,5,6,9,7,8] => [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,4,5,7,6,8,9,1] => [1,2,3,4,5,7,8,9,6] => [.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,5,8,6,7,9,1] => [1,2,3,4,5,8,9,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,5,9,6,7,8,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,4,5,9,6,8,7,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,4,5,8,7,6,9,1] => [1,2,3,4,5,8,9,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,4,5,9,7,6,8,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,9,7,8,6,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,6,5,7,8,9,1] => [1,2,3,4,6,7,8,9,5] => [.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,4,9,5,6,7,8,1] => [1,2,3,4,9,5,6,7,8] => [.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,4,9,6,5,7,8,1] => [1,2,3,4,9,5,6,7,8] => [.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,5,1] => [1,2,3,4,9,5,6,7,8] => [.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,3,9,4,5,6,7,8,1] => [1,2,3,9,4,5,6,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,3,9,5,4,6,7,8,1] => [1,2,3,9,4,5,6,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,3,9,5,6,7,8,4,1] => [1,2,3,9,4,5,6,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,8,9,1] => [1,2,4,5,6,7,8,9,3] => [.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> ? = 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,9,3,4,5,6,7,8,1] => [1,2,9,3,4,5,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [2,9,4,3,5,6,7,8,1] => [1,2,9,3,4,5,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [2,9,4,5,6,7,8,3,1] => [1,2,9,3,4,5,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [1,3,4,5,6,7,8,9,2] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ? = 1
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,2,9,4,5,6,7,8,1] => [1,3,9,2,4,5,6,7,8] => [.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,9,1] => [1,4,5,6,7,8,9,2,3] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [7,2,3,4,5,6,9,8,1] => [1,7,9,2,3,4,5,6,8] => [.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,2,3,4,5,6,7,9,1] => [1,8,9,2,3,4,5,6,7] => [.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,2,3,4,5,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [9,2,4,5,6,8,7,3,1] => [1,9,2,3,4,5,6,8,7] => [.,[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 2
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [9,2,7,6,5,4,3,8,1] => [1,9,2,3,7,4,6,5,8] => [.,[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]]
=> ? = 3
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,8,9,1] => [1,4,5,6,7,8,9,2,3] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? = 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,3,2,4,5,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [9,3,2,5,6,7,8,4,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [9,3,4,2,5,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [9,3,4,5,2,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
Description
The number of maximal left branches of a binary tree.
A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.
Matching statistic: St000568
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,3,6,5,7,2,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 1 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,4,3,6,5,7,8,2] => [1,2,4,6,7,8,3,5] => ?
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,6,5,7,1,8] => [1,2,4,6,7,3,5,8] => ?
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [2,4,3,6,5,7,8,1] => [1,2,4,6,7,8,3,5] => ?
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,0,1,1,1,0,0,0]
=> [3,2,4,5,1,8,7,6] => [1,3,4,5,2,6,8,7] => ?
=> ? = 2 + 1
[1,1,1,0,1,0,0,1,0,1,1,1,0,0,0,0]
=> [4,2,3,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> [5,2,3,4,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1 + 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0,1,0]
=> [5,2,4,3,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1 + 1
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [4,3,2,5,8,7,6,1] => [1,4,5,8,2,3,6,7] => ?
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [5,3,2,4,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0,1,0]
=> [5,3,4,2,7,6,1,8] => [1,5,7,2,3,4,6,8] => ?
=> ? = 1 + 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> [7,4,3,6,5,2,8,1] => [1,7,8,2,4,6,3,5] => ?
=> ? = 2 + 1
[1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [8,5,3,4,7,6,2,1] => [1,8,2,5,7,3,4,6] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,0,1,1,0,0,0,0,0]
=> [8,5,4,3,7,6,2,1] => [1,8,2,5,7,3,4,6] => ?
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,5,6,7,9,8,1] => [1,2,3,4,5,6,7,9,8] => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,4,5,6,8,7,9,1] => [1,2,3,4,5,6,8,9,7] => [.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,4,5,6,9,7,8,1] => [1,2,3,4,5,6,9,7,8] => [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,4,5,6,9,8,7,1] => [1,2,3,4,5,6,9,7,8] => [.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [2,3,4,5,7,6,8,9,1] => [1,2,3,4,5,7,8,9,6] => [.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [2,3,4,5,8,6,7,9,1] => [1,2,3,4,5,8,9,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [2,3,4,5,9,6,7,8,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [2,3,4,5,9,6,8,7,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,4,5,8,7,6,9,1] => [1,2,3,4,5,8,9,6,7] => [.,[.,[.,[.,[.,[[.,[.,.]],[.,.]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,4,5,9,7,6,8,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,4,5,9,7,8,6,1] => [1,2,3,4,5,9,6,7,8] => [.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,6,5,7,8,9,1] => [1,2,3,4,6,7,8,9,5] => [.,[.,[.,[.,[[.,[.,[.,[.,.]]]],.]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [2,3,4,9,5,6,7,8,1] => [1,2,3,4,9,5,6,7,8] => [.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [2,3,4,9,6,5,7,8,1] => [1,2,3,4,9,5,6,7,8] => [.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,9,6,7,8,5,1] => [1,2,3,4,9,5,6,7,8] => [.,[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,3,9,4,5,6,7,8,1] => [1,2,3,9,4,5,6,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,3,9,5,4,6,7,8,1] => [1,2,3,9,4,5,6,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [2,3,9,5,6,7,8,4,1] => [1,2,3,9,4,5,6,7,8] => [.,[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,3,5,6,7,8,9,1] => [1,2,4,5,6,7,8,9,3] => [.,[.,[[.,[.,[.,[.,[.,[.,.]]]]]],.]]]
=> ? = 1 + 1
[1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [2,9,3,4,5,6,7,8,1] => [1,2,9,3,4,5,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [2,9,4,3,5,6,7,8,1] => [1,2,9,3,4,5,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [2,9,4,5,6,7,8,3,1] => [1,2,9,3,4,5,6,7,8] => [.,[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [1,3,4,5,6,7,8,9,2] => [.,[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]]
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [3,2,9,4,5,6,7,8,1] => [1,3,9,2,4,5,6,7,8] => [.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,9,1] => [1,4,5,6,7,8,9,2,3] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [7,2,3,4,5,6,9,8,1] => [1,7,9,2,3,4,5,6,8] => [.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [8,2,3,4,5,6,7,9,1] => [1,8,9,2,3,4,5,6,7] => [.,[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,2,3,4,5,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [9,2,4,5,6,8,7,3,1] => [1,9,2,3,4,5,6,8,7] => [.,[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 2 + 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [9,2,7,6,5,4,3,8,1] => [1,9,2,3,7,4,6,5,8] => [.,[[.,.],[.,[[.,.],[[.,.],[.,.]]]]]]
=> ? = 3 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,8,9,1] => [1,4,5,6,7,8,9,2,3] => [.,[[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]]
=> ? = 1 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [9,3,2,4,5,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [9,3,2,5,6,7,8,4,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [9,3,4,2,5,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0,0]
=> [9,3,4,5,2,6,7,8,1] => [1,9,2,3,4,5,6,7,8] => [.,[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1 + 1
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
Matching statistic: St000884
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [5,2,3,4,1,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [5,2,3,4,1,7,6] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [5,2,3,4,6,1,7] => [1,5,6,2,3,4,7] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,2,3,4,5,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,5,4,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [7,2,3,6,5,4,1] => [1,7,2,3,4,6,5] => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [5,2,4,3,1,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [5,2,4,3,1,7,6] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [5,2,4,3,6,1,7] => [1,5,6,2,3,4,7] => ? = 1
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [6,2,4,3,5,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,4,5,3,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,2,4,6,5,3,1] => [1,7,2,3,4,6,5] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,2,5,4,3,1,7] => [1,6,2,3,5,4,7] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => [1,6,7,2,3,5,4] => ? = 2
[1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [7,2,5,4,3,6,1] => [1,7,2,3,5,4,6] => ? = 2
[1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [7,2,5,4,6,3,1] => [1,7,2,3,5,6,4] => ? = 2
[1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [7,2,6,4,5,3,1] => [1,7,2,3,6,4,5] => ? = 2
[1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [7,2,6,5,4,3,1] => [1,7,2,3,6,4,5] => ? = 2
[1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [5,3,2,4,1,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [5,3,2,4,1,7,6] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [5,3,2,4,6,1,7] => [1,5,6,2,3,4,7] => ? = 1
[1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,3,2,5,4,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [7,3,2,6,5,4,1] => [1,7,2,3,4,6,5] => ? = 2
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,3,4,2,1,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [5,3,4,2,1,7,6] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,3,4,2,6,1,7] => [1,5,6,2,3,4,7] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,3,4,2,5,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,3,4,5,2,1,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,3,4,6,5,2,1] => [1,7,2,3,4,6,5] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,5,4,2,1,7] => [1,6,2,3,5,4,7] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,3,5,4,2,7,1] => [1,6,7,2,3,5,4] => ? = 2
[1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [7,3,5,4,2,6,1] => [1,7,2,3,5,4,6] => ? = 2
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [7,3,5,4,6,2,1] => [1,7,2,3,5,6,4] => ? = 2
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [7,3,6,4,5,2,1] => [1,7,2,3,6,4,5] => ? = 2
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [7,3,6,5,4,2,1] => [1,7,2,3,6,4,5] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [1,5,2,4,3,6,7] => ? = 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [1,5,2,4,3,6,7] => ? = 2
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,4,3,2,6,1,7] => [1,5,6,2,4,3,7] => ? = 2
[1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => [1,5,6,7,2,4,3] => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,4,3,2,7,6,1] => [1,5,7,2,4,3,6] => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,4,3,2,5,1,7] => [1,6,2,4,3,5,7] => ? = 2
[1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => [1,6,7,2,4,3,5] => ? = 2
[1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [7,4,3,2,5,6,1] => [1,7,2,4,3,5,6] => ? = 2
[1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [7,4,3,2,6,5,1] => [1,7,2,4,3,5,6] => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,4,3,5,2,1,7] => [1,6,2,4,5,3,7] => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [1,6,7,2,4,5,3] => ? = 2
[1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [7,4,3,5,2,6,1] => [1,7,2,4,5,3,6] => ? = 2
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => ? = 2
[1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [7,4,3,6,5,2,1] => [1,7,2,4,6,3,5] => ? = 2
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St000994
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,2,4,5,1,6,7] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 1
[1,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [3,2,4,5,1,7,6] => [1,3,4,5,2,6,7] => [1,5,3,4,2,6,7] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [3,2,4,5,6,1,7] => [1,3,4,5,6,2,7] => [1,6,3,4,5,2,7] => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => [1,3,4,5,6,7,2] => [1,7,3,4,5,6,2] => ? = 1
[1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [3,2,4,5,7,6,1] => [1,3,4,5,7,2,6] => [1,7,3,4,5,2,6] => ? = 1
[1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [3,2,4,6,5,1,7] => [1,3,4,6,2,5,7] => [1,6,3,4,2,5,7] => ? = 1
[1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [3,2,4,6,5,7,1] => [1,3,4,6,7,2,5] => [1,7,3,4,2,6,5] => ? = 1
[1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [3,2,4,7,5,6,1] => [1,3,4,7,2,5,6] => [1,7,3,4,2,5,6] => ? = 1
[1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,2,4,7,6,5,1] => [1,3,4,7,2,5,6] => [1,7,3,4,2,5,6] => ? = 1
[1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,2,5,4,1,6,7] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 1
[1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [3,2,5,4,1,7,6] => [1,3,5,2,4,6,7] => [1,5,3,2,4,6,7] => ? = 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [3,2,5,4,6,1,7] => [1,3,5,6,2,4,7] => [1,6,3,2,5,4,7] => ? = 1
[1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [3,2,5,4,6,7,1] => [1,3,5,6,7,2,4] => [1,7,3,2,5,6,4] => ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,7,6,1] => [1,3,5,7,2,4,6] => [1,7,3,2,5,4,6] => ? = 1
[1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [3,2,6,4,5,1,7] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [3,2,6,4,5,7,1] => [1,3,6,7,2,4,5] => [1,7,3,2,4,6,5] => ? = 1
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [3,2,7,4,5,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [3,2,7,4,6,5,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => [1,3,6,2,4,5,7] => [1,6,3,2,4,5,7] => ? = 1
[1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [3,2,6,5,4,7,1] => [1,3,6,7,2,4,5] => [1,7,3,2,4,6,5] => ? = 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> [3,2,7,5,4,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [3,2,7,5,6,4,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,2,7,6,5,4,1] => [1,3,7,2,4,6,5] => [1,6,3,2,7,4,5] => ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,2,3,5,1,6,7] => [1,4,5,2,3,6,7] => [1,5,2,4,3,6,7] => ? = 1
[1,1,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,2,3,5,1,7,6] => [1,4,5,2,3,6,7] => [1,5,2,4,3,6,7] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,2,3,5,6,1,7] => [1,4,5,6,2,3,7] => [1,6,2,4,5,3,7] => ? = 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => [1,4,5,6,7,2,3] => [1,7,2,4,5,6,3] => ? = 1
[1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [4,2,3,5,7,6,1] => [1,4,5,7,2,3,6] => [1,7,2,4,5,3,6] => ? = 1
[1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [4,2,3,6,5,1,7] => [1,4,6,2,3,5,7] => [1,6,2,4,3,5,7] => ? = 1
[1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [4,2,3,6,5,7,1] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[1,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [4,2,3,7,5,6,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,2,3,7,6,5,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [5,2,3,4,1,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,0,1,0,0,0,1,1,0,0]
=> [5,2,3,4,1,7,6] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [5,2,3,4,6,1,7] => [1,5,6,2,3,4,7] => [1,6,2,3,5,4,7] => ? = 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => [1,7,2,3,5,6,4] => ? = 1
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [5,2,3,4,7,6,1] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,2,3,4,5,1,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [1,6,7,2,3,4,5] => [1,7,2,3,4,6,5] => ? = 1
[1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,5,4,1,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,2,3,5,4,7,1] => [1,6,7,2,3,4,5] => [1,7,2,3,4,6,5] => ? = 1
[1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [5,2,4,3,1,6,7] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [5,2,4,3,1,7,6] => [1,5,2,3,4,6,7] => [1,5,2,3,4,6,7] => ? = 1
[1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [5,2,4,3,6,1,7] => [1,5,6,2,3,4,7] => [1,6,2,3,5,4,7] => ? = 1
[1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => [1,5,6,7,2,3,4] => [1,7,2,3,5,6,4] => ? = 1
[1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> [5,2,4,3,7,6,1] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [6,2,4,3,5,1,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [6,2,4,3,5,7,1] => [1,6,7,2,3,4,5] => [1,7,2,3,4,6,5] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,2,4,5,3,1,7] => [1,6,2,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 1
[1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,2,4,5,3,7,1] => [1,6,7,2,3,4,5] => [1,7,2,3,4,6,5] => ? = 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
The following 115 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000996The number of exclusive left-to-right maxima of a permutation. St000035The number of left outer peaks of a permutation. St000703The number of deficiencies of a permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000619The number of cyclic descents of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001729The number of visible descents of a permutation. St000354The number of recoils of a permutation. St001469The holeyness of a permutation. St001665The number of pure excedances of a permutation. St000353The number of inner valleys of a permutation. St000632The jump number of the poset. St000455The second largest eigenvalue of a graph if it is integral. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000098The chromatic number of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St001330The hat guessing number of a graph. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000092The number of outer peaks of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001331The size of the minimal feedback vertex set. St001358The largest degree of a regular subgraph of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001638The book thickness of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000544The cop number of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001746The coalition number of a graph. St000259The diameter of a connected graph. St001057The Grundy value of the game of creating an independent set in a graph. St001674The number of vertices of the largest induced star graph in the graph. St001427The number of descents of a signed permutation. St000454The largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000307The number of rowmotion orbits of a poset. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000640The rank of the largest boolean interval in a poset. St000822The Hadwiger number of the graph. St001734The lettericity of a graph. St001642The Prague dimension of a graph. St001323The independence gap of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001352The number of internal nodes in the modular decomposition of a graph. St001597The Frobenius rank of a skew partition. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001960The number of descents of a permutation minus one if its first entry is not one. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000264The girth of a graph, which is not a tree. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000252The number of nodes of degree 3 of a binary tree. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000711The number of big exceedences of a permutation. St000805The number of peaks of the associated bargraph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001728The number of invisible descents of a permutation.
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