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Your data matches 24 different statistics following compositions of up to 3 maps.
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Matching statistic: St000157
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => [[1,2],[3]]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,3,2,5] => [[1,2,5],[3],[4]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,4,2,5,3] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
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 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00130: Permutations —descent tops⟶ 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
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => 0 => 0
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => 01 => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 01 => 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,2,3] => 00 => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,2,3] => 001 => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,3,2,4] => 010 => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => 001 => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => 001 => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,2,3] => 001 => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => 011 => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,3,2,4] => 010 => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,3,4] => 000 => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 011 => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => 010 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => 000 => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => 0011 => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,4,2,5,3] => 0011 => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => 0001 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,4,2,3,5] => 0010 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,3,5,2,4] => 0001 => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,5,2,4,3] => 0011 => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,3,2,5] => 0110 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => 0001 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => 0001 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,4,5] => 0100 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => 0001 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => 0001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => 0001 => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,2,4,3] => 0011 => 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,2,5,3] => 0011 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => 0001 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,2,3,5] => 0010 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => 0001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,5,3,2,4] => 0101 => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,4,2,5,3] => 0011 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,2,3] => 0011 => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => 0010 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,5,4,2,3] => 0011 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => 0010 => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => 0010 => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => 0000 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => 0101 => 2
[1,1,1,1,0,1,0,1,0,0,1,0,0,0,1,0]
=> [8,5,3,2,1,4,6,7] => [1,8,7,6,4,2,5,3] => ? => ? = 5
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [8,7,2,3,1,4,5,6] => [1,8,6,4,3,2,7,5] => ? => ? = 5
[1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [8,5,3,1,2,4,6,7] => [1,8,7,6,4,2,5,3] => ? => ? = 5
[1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,5,2,1,3,4,6,7] => [1,8,7,6,4,2,5,3] => ? => ? = 5
[1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => [1,8,7,6,4,2,5,3] => ? => ? = 5
[]
=> [] => [] => => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [9,1,2,3,4,5,6,7,8,10] => [1,9,8,7,6,5,4,3,2,10] => 011111110 => ? = 7
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0,0]
=> [9,2,1,3,4,5,6,7,8,10] => [1,9,8,7,6,5,4,3,2,10] => 011111110 => ? = 7
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000996
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,1] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [3,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,2,3] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [3,1,4,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,3,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [3,1,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => [2,4,1,3] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => [3,1,4,2,6,5,8,7] => [3,1,4,2,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [2,1,5,3,6,4,8,7] => [2,1,5,3,6,4,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => [3,1,5,2,6,4,8,7] => [3,1,5,2,6,4,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,8,7] => [4,1,5,2,6,3,8,7] => [4,1,5,2,6,3,7] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,8,5,7] => [2,1,4,3,7,5,8,6] => [2,1,4,3,7,5,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => [3,1,4,2,7,5,8,6] => [3,1,4,2,7,5,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,8,5,7] => [2,1,5,3,7,4,8,6] => [2,1,5,3,7,4,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => [3,1,5,2,7,4,8,6] => [3,1,5,2,7,4,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => [2,1,6,3,7,4,8,5] => [2,1,6,3,7,4,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,8,3,5,7] => [3,1,6,2,7,4,8,5] => [3,1,6,2,7,4,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => [4,1,6,2,7,3,8,5] => [4,1,6,2,7,3,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [5,1,6,2,7,3,8,4] => [5,1,6,2,7,3,4] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => [2,1,4,3,6,5,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7,8] => [3,1,4,2,6,5,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7,8] => [2,1,5,3,6,4,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7,8] => [3,1,5,2,6,4,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7,8] => [4,1,5,2,6,3,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5,8] => [2,1,4,3,7,5,6,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5,8] => [2,1,6,3,7,4,5,8] => [2,1,6,3,7,4,5] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => [2,1,4,3,8,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,3,7,8,5] => [2,1,5,3,8,4,6,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => [3,1,5,2,8,4,6,7] => [3,1,5,2,4,6,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,6,1,3,7,8,5] => [4,1,5,2,8,3,6,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,6,7,3,8,5] => [3,1,6,2,8,4,5,7] => [3,1,6,2,4,5,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => [5,1,6,2,8,3,4,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,6,7,8,3,5] => [2,1,7,3,8,4,5,6] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,6,7,8,3,5] => [3,1,7,2,8,4,5,6] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => [4,1,7,2,8,3,5,6] => [4,1,7,2,3,5,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,6,7,1,8,3,5] => [5,1,7,2,8,3,4,6] => [5,1,7,2,3,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,6,7,8,1,3,5] => [6,1,7,2,8,3,4,5] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,8,6] => [2,1,4,3,6,8,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,8,6] => [3,1,4,2,6,8,5,7] => [3,1,4,2,6,5,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,7,8,5,6] => [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,7,8,5,6] => [3,1,4,2,7,8,5,6] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,4,7,1,3,8,5,6] => [4,1,5,2,7,8,3,6] => [4,1,5,2,7,3,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,7,8,3,5,6] => [2,1,6,3,7,8,4,5] => [2,1,6,3,7,4,5] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4,7,1,8,3,5,6] => [4,1,6,2,7,8,3,5] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => [5,1,6,2,7,8,3,4] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6,8] => [2,1,4,3,6,7,5,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6,8] => [3,1,4,2,6,7,5,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => [2,1,5,3,6,7,4,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6,8] => [4,1,5,2,6,7,3,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6,8] => [2,1,4,3,5,7,6,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6,8] => [3,1,4,2,5,7,6,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => [3,1,5,2,4,7,6,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6,8] => [4,1,6,2,3,7,5,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => [2,1,4,3,5,8,6,7] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,7,8,6] => [2,1,5,3,4,8,6,7] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,5,1,3,7,8,6] => [4,1,5,2,3,8,6,7] => ? => ? = 2
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000204
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000204: Binary trees ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1] => [.,.]
=> 0
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [[.,.],.]
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [.,[.,.]]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> 0
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[.,.],[[.,.],.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 0
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,8,5,7] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [[.,.],[[.,[.,.]],[.,[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,8,5,7] => [[.,.],[[.,[.,.]],[[.,.],[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => [[.,.],[[.,[.,[.,.]]],[.,[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => [[.,[.,.]],[[.,.],[.,[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => [[.,[.,.]],[[.,.],[[.,.],[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [2,4,6,1,3,8,5,7] => [[.,[.,[.,.]]],[.,[[.,.],[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => [[.,[.,[.,.]]],[[.,.],[.,[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,2,4,3,6,5,8,7] => [.,[.,[[.,.],[[.,.],[[.,.],.]]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,2,4,3,6,8,5,7] => ?
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5,8,7] => [.,[.,[[.,[.,.]],[.,[[.,.],.]]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,2,4,6,3,8,5,7] => [.,[.,[[.,[.,.]],[[.,.],[.,.]]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,2,4,6,8,3,5,7] => [.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,4,2,3,6,5,8,7] => [.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,4,6,2,3,5,8,7] => [.,[[.,[.,.]],[.,[.,[[.,.],.]]]]]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,1,2,3,6,5,8,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [4,1,2,6,3,5,8,7] => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [[.,.],[.,[[.,.],[[.,.],[.,.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0,1,0]
=> [4,1,2,6,8,3,5,7] => ?
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,1,6,2,3,8,5,7] => [[.,.],[[.,.],[.,[[.,.],[.,.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,0,1,0]
=> [4,1,6,8,2,3,5,7] => [[.,.],[[.,[.,.]],[.,[.,[.,.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,1,0,0]
=> [4,6,1,2,3,5,8,7] => [[.,[.,.]],[.,[.,[.,[[.,.],.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> [4,6,1,2,3,8,5,7] => [[.,[.,.]],[.,[.,[[.,.],[.,.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> [4,6,1,2,8,3,5,7] => [[.,[.,.]],[.,[[.,.],[.,[.,.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,0,1,0,0]
=> [4,6,1,8,2,3,5,7] => [[.,[.,.]],[[.,.],[.,[.,[.,.]]]]]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0]
=> [4,6,8,1,2,3,5,7] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,4,2,6,5,8,7] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [3,1,4,2,6,8,5,7] => [[.,.],[[.,.],[[.,[.,.]],[.,.]]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [3,4,1,2,6,5,8,7] => [[.,[.,.]],[.,[[.,.],[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [3,4,1,2,6,8,5,7] => [[.,[.,.]],[.,[[.,[.,.]],[.,.]]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [3,4,1,6,8,2,5,7] => [[.,[.,.]],[[.,[.,.]],[.,[.,.]]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5,8,7] => [[.,[.,[.,.]]],[.,[.,[[.,.],.]]]]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,4,6,1,8,2,5,7] => ?
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,6,8,1,2,5,7] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,3,4,2,6,5,8,7] => [.,[[.,[.,.]],[[.,.],[[.,.],.]]]]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,3,4,2,6,8,5,7] => ?
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,3,4,6,2,5,8,7] => [.,[[.,[.,[.,.]]],[.,[[.,.],.]]]]
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,3,4,6,8,2,5,7] => [.,[[.,[.,[.,[.,.]]]],[.,[.,.]]]]
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,3,2,4,6,5,8,7] => [.,[[.,.],[.,[[.,.],[[.,.],.]]]]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,3,2,4,6,8,5,7] => [.,[[.,.],[.,[[.,[.,.]],[.,.]]]]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,3,2,6,4,8,5,7] => [.,[[.,.],[[.,.],[[.,.],[.,.]]]]]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,3,6,2,8,4,5,7] => ?
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [3,1,2,4,6,5,8,7] => [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [3,1,2,6,4,5,8,7] => [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [3,1,2,6,8,4,5,7] => [[.,.],[.,[[.,[.,.]],[.,[.,.]]]]]
=> ? = 2
Description
The number of internal nodes of a binary tree.
That is, the total number of nodes of the tree minus [[St000203]]. A counting formula for the total number of internal nodes across all binary trees of size $n$ is given in [1]. This is equivalent to the number of internal triangles in all triangulations of an $(n+1)$-gon.
Matching statistic: St000155
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => [] => ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => [1] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1] => 0
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => [1,2] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,1] => 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [1,2] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2] => 0
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [3,1,2] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [3,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,2,3] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,3,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [3,1,4,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,3,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [3,1,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [4,1,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [2,4,1,3,5] => [2,4,1,3] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [3,4,1,2,5] => [3,4,1,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [3,5,1,2,4] => [3,1,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [4,5,1,2,3] => [4,1,2,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [2,4,1,5,3] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [3,4,1,5,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [2,3,1,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,4,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,4,5,1,3] => [2,4,1,3] => 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => [3,1,4,2,6,5,8,7] => [3,1,4,2,6,5,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [2,1,5,3,6,4,8,7] => [2,1,5,3,6,4,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => [3,1,5,2,6,4,8,7] => [3,1,5,2,6,4,7] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,8,7] => [4,1,5,2,6,3,8,7] => [4,1,5,2,6,3,7] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,8,5,7] => [2,1,4,3,7,5,8,6] => [2,1,4,3,7,5,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => [3,1,4,2,7,5,8,6] => [3,1,4,2,7,5,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,8,5,7] => [2,1,5,3,7,4,8,6] => [2,1,5,3,7,4,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => [3,1,5,2,7,4,8,6] => [3,1,5,2,7,4,6] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => [2,1,6,3,7,4,8,5] => [2,1,6,3,7,4,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,8,3,5,7] => [3,1,6,2,7,4,8,5] => [3,1,6,2,7,4,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => [4,1,6,2,7,3,8,5] => [4,1,6,2,7,3,5] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [5,1,6,2,7,3,8,4] => [5,1,6,2,7,3,4] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => [2,1,4,3,6,5,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7,8] => [3,1,4,2,6,5,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7,8] => [2,1,5,3,6,4,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7,8] => [3,1,5,2,6,4,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7,8] => [4,1,5,2,6,3,7,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5,8] => [2,1,4,3,7,5,6,8] => ? => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5,8] => [2,1,6,3,7,4,5,8] => [2,1,6,3,7,4,5] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => [2,1,4,3,8,5,6,7] => [2,1,4,3,5,6,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,3,7,8,5] => [2,1,5,3,8,4,6,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => [3,1,5,2,8,4,6,7] => [3,1,5,2,4,6,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,6,1,3,7,8,5] => [4,1,5,2,8,3,6,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,6,7,3,8,5] => [3,1,6,2,8,4,5,7] => [3,1,6,2,4,5,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => [5,1,6,2,8,3,4,7] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,6,7,8,3,5] => [2,1,7,3,8,4,5,6] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,6,7,8,3,5] => [3,1,7,2,8,4,5,6] => ? => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => [4,1,7,2,8,3,5,6] => [4,1,7,2,3,5,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,6,7,1,8,3,5] => [5,1,7,2,8,3,4,6] => [5,1,7,2,3,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,6,7,8,1,3,5] => [6,1,7,2,8,3,4,5] => ? => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,8,6] => [2,1,4,3,6,8,5,7] => [2,1,4,3,6,5,7] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,8,6] => [3,1,4,2,6,8,5,7] => [3,1,4,2,6,5,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,7,8,5,6] => [2,1,4,3,7,8,5,6] => [2,1,4,3,7,5,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,7,8,5,6] => [3,1,4,2,7,8,5,6] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,4,7,1,3,8,5,6] => [4,1,5,2,7,8,3,6] => [4,1,5,2,7,3,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,7,8,3,5,6] => [2,1,6,3,7,8,4,5] => [2,1,6,3,7,4,5] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4,7,1,8,3,5,6] => [4,1,6,2,7,8,3,5] => ? => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => [5,1,6,2,7,8,3,4] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6,8] => [2,1,4,3,6,7,5,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6,8] => [3,1,4,2,6,7,5,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => [2,1,5,3,6,7,4,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6,8] => [4,1,5,2,6,7,3,8] => ? => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6,8] => [2,1,4,3,5,7,6,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6,8] => [3,1,4,2,5,7,6,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => [3,1,5,2,4,7,6,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6,8] => [4,1,6,2,3,7,5,8] => ? => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => [2,1,4,3,5,8,6,7] => ? => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,7,8,6] => [2,1,5,3,4,8,6,7] => ? => ? = 2
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
Matching statistic: St001499
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,8,7] => [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,8,7] => [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,8,7] => [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,8,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,8,5,7] => [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,6,8,5,7] => [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,4,6,3,8,5,7] => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,6,3,8,5,7] => [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,4,6,8,3,5,7] => [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,4,1,6,8,3,5,7] => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,4,6,1,8,3,5,7] => [1,1,0,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,6,8,1,3,5,7] => [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,6,5,7,8] => [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,6,3,5,7,8] => [1,1,0,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [2,4,1,6,3,5,7,8] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,4,6,1,3,5,7,8] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5,8] => [1,1,0,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,7,3,5,8] => [1,1,0,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,6,7,8,5] => [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [2,1,4,6,3,7,8,5] => [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,4,1,6,3,7,8,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [2,4,6,1,3,7,8,5] => [1,1,0,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,4,1,6,7,3,8,5] => [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,4,6,7,1,3,8,5] => [1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,4,6,7,8,3,5] => [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,4,1,6,7,8,3,5] => [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,4,6,1,7,8,3,5] => [1,1,0,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,4,6,7,1,8,3,5] => [1,1,0,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,6,7,8,1,3,5] => [1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,8,6] => [1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,8,6] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,7,8,5,6] => [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,4,1,3,7,8,5,6] => [1,1,0,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,4,7,1,3,8,5,6] => [1,1,0,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,7,8,3,5,6] => [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4,7,1,8,3,5,6] => [1,1,0,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,7,8,1,3,5,6] => [1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,7,5,6,8] => ?
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,4,1,3,7,5,6,8] => ?
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,4,7,3,5,6,8] => ?
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,4,7,1,3,5,6,8] => ?
=> ?
=> ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [2,1,4,3,5,7,6,8] => ?
=> ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,1,3,5,7,6,8] => ?
=> ?
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,1,5,3,7,6,8] => [1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,7,3,6,8] => [1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,1,4,3,5,7,8,6] => [1,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,1,4,5,3,7,8,6] => ?
=> ?
=> ? = 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
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [1,4,2,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,4,5,3,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,3,5,2] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,5,2,4,3] => [1,4,5,3,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,3,2,5] => [1,4,2,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,3,5,4,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,2,4,3] => [1,4,5,3,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [1,4,3,5,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,5,3,2,4] => [1,4,2,5,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,2,3] => [1,3,5,2,4] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,5,4,2,3] => [1,3,5,2,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,6,2,7,3,5,4] => [1,6,5,7,4,2,3] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,2,5,3,6,4] => [1,5,6,7,3,4,2] => ? = 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [1,6,2,5,3,7,4] => [1,5,6,7,3,2,4] => ? = 3
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,5,3,7,2,6,4] => [1,6,5,7,3,4,2] => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => [1,7,2,6,3,4,5] => [1,6,4,5,7,3,2] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => [1,6,2,7,3,4,5] => [1,6,4,5,7,2,3] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => [1,7,2,5,3,4,6] => [1,5,4,6,3,7,2] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => [1,6,2,5,3,4,7] => [1,5,4,6,3,2,7] => ? = 2
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,5,3,4,7,2,6] => [1,6,4,5,3,7,2] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,4,7,2,5,3,6] => [1,5,6,4,3,7,2] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,6,2,7,3,5,4] => [1,6,5,7,4,2,3] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,2,5,4,3,6] => [1,5,6,3,4,7,2] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [1,6,2,5,4,3,7] => [1,5,6,3,4,2,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,5,4,3,7,2,6] => [1,6,5,3,4,7,2] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => [1,7,2,6,3,4,5] => [1,6,4,5,7,3,2] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,5,2,1] => [1,6,2,7,3,4,5] => [1,6,4,5,7,2,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [1,6,2,5,7,3,4] => [1,5,4,7,6,2,3] => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [1,5,7,2,6,3,4] => [1,6,4,7,5,3,2] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,4,3,6,2,5,7] => [1,5,4,3,6,2,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => [1,7,2,6,3,4,5] => [1,6,4,5,7,3,2] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => [1,6,2,7,3,4,5] => [1,6,4,5,7,2,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,6,2,5,7,3,4] => [1,5,4,7,6,2,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,5,7,2,6,3,4] => [1,6,4,7,5,3,2] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => [1,4,7,5,6,3,2] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,6,2,4,5,7,3] => [1,4,7,5,6,2,3] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,4,6,2,5,7,3] => [1,5,7,4,6,2,3] => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,3,5,2,7,4] => [1,6,5,7,2,3,4] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,7,2,5,3,6,4] => [1,5,6,7,3,4,2] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,2,3,1] => [1,7,2,6,3,4,5] => [1,6,4,5,7,3,2] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,6,3,4,5,2,7] => [1,6,4,5,2,3,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,2,3,1] => [1,7,2,5,3,4,6] => [1,5,4,6,3,7,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [1,6,3,4,7,2,5] => [1,5,4,6,7,3,2] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,2,3,1] => [1,5,2,6,3,4,7] => [1,5,4,6,2,3,7] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [1,4,7,2,5,3,6] => [1,5,6,4,3,7,2] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [1,7,2,6,4,3,5] => [1,6,5,3,7,4,2] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [1,6,4,3,5,2,7] => [1,6,5,3,2,4,7] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [1,7,2,5,3,6,4] => [1,5,6,7,3,4,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [1,6,4,3,7,2,5] => [1,5,6,3,7,4,2] => ? = 3
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [1,5,2,6,4,3,7] => [1,5,6,3,2,4,7] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,5,1] => [1,7,2,6,5,3,4] => [1,6,4,7,3,5,2] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [1,6,5,2,7,3,4] => [1,6,4,7,2,5,3] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => [1,7,2,5,3,4,6] => [1,5,4,6,3,7,2] => ? = 2
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [1,6,7,2,5,3,4] => [1,5,4,7,3,6,2] => ? = 2
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [1,5,2,6,7,3,4] => [1,5,4,7,2,6,3] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [1,4,3,6,7,2,5] => [1,5,4,3,7,6,2] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,5,1] => [1,7,2,6,5,3,4] => [1,6,4,7,3,5,2] => ? = 3
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [1,6,5,2,7,3,4] => [1,6,4,7,2,5,3] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,6,1] => [1,7,2,5,3,4,6] => [1,5,4,6,3,7,2] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [1,6,7,2,5,3,4] => [1,5,4,7,3,6,2] => ? = 2
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [1,5,2,6,7,3,4] => [1,5,4,7,2,6,3] => ? = 2
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]].
Matching statistic: St000354
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 56%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 29% ●values known / values provided: 29%●distinct values known / distinct values provided: 56%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 0
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,3,5,4,2] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,4,2,5,3] => [1,3,5,2,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,5,2,4,3] => [1,3,5,4,2] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,2,4,3] => [1,3,5,4,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,2,5,3] => [1,3,5,2,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,5,3,2,4] => [1,4,3,5,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,4,2,5,3] => [1,3,5,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,2,3] => [1,4,5,3,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,5,4,2,3] => [1,4,5,3,2] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,3,5,2] => 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,5,3,7,2,6,4] => [1,5,3,7,2,6,4] => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,5,3,4,7,2,6] => [1,6,3,4,2,7,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => [1,5,3,6,2,4,7] => [1,5,3,6,2,4,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,5,4,3,7,2,6] => [1,6,4,3,2,7,5] => ? = 3
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,7,2,1] => [1,5,7,2,6,3,4] => [1,4,6,7,2,5,3] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,4,3,6,2,5,7] => [1,5,3,2,6,4,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,7,2,1] => [1,5,7,2,6,3,4] => [1,4,6,7,2,5,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [1,4,7,5,2,6,3] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,4,6,2,5,7,3] => [1,4,7,2,5,3,6] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,3,5,2,7,4] => [1,5,3,7,4,2,6] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [1,6,3,7,2,5,4] => [1,5,3,7,6,2,4] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,6,3,4,5,2,7] => [1,6,3,4,5,2,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [1,6,3,4,7,2,5] => [1,6,3,4,7,2,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,2,3,1] => [1,6,3,5,2,4,7] => [1,5,3,6,4,2,7] => ? = 2
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [1,6,4,3,5,2,7] => [1,6,4,3,5,2,7] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [1,6,4,3,7,2,5] => [1,6,4,3,7,2,5] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [1,6,5,2,7,3,4] => [1,4,6,7,3,2,5] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,7,1] => [1,6,7,2,4,3,5] => [1,4,6,5,7,2,3] => ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [1,4,3,6,7,2,5] => [1,6,3,2,7,4,5] => ? = 2
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [1,6,5,2,7,3,4] => [1,4,6,7,3,2,5] => ? = 3
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,2,7,1] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? = 2
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [1,6,7,2,4,5,3] => [1,4,7,5,6,2,3] => ? = 2
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [1,4,6,7,2,5,3] => [1,5,7,2,6,3,4] => ? = 2
[1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,2,7,1] => [1,4,6,7,2,3,5] => [1,5,6,2,7,3,4] => ? = 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => [1,5,3,7,2,6,4] => [1,5,3,7,2,6,4] => ? = 3
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [6,7,4,2,3,5,1] => [1,6,5,3,4,2,7] => [1,6,4,5,3,2,7] => ? = 3
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,7,1] => [1,6,7,2,5,3,4] => [1,4,6,7,5,2,3] => ? = 2
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,7,1] => [1,5,3,4,2,6,7] => [1,5,3,4,2,6,7] => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [6,4,5,2,3,7,1] => [1,6,7,2,4,3,5] => [1,4,6,5,7,2,3] => ? = 2
[1,0,1,1,1,0,0,1,1,0,1,0,0,0]
=> [5,4,6,2,3,7,1] => [1,5,3,6,7,2,4] => [1,6,3,7,2,4,5] => ? = 2
[1,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,7,3,2,4,5,1] => [1,6,5,4,2,7,3] => [1,5,7,4,3,2,6] => ? = 4
[1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,4,7,1] => [1,6,7,2,5,4,3] => [1,4,7,6,5,2,3] => ? = 3
[1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [5,6,3,2,4,7,1] => [1,5,4,2,6,7,3] => [1,4,7,3,2,5,6] => ? = 3
[1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,5,7,1] => [1,6,7,2,4,3,5] => [1,4,6,5,7,2,3] => ? = 2
[1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,6,7,1] => [1,5,6,7,2,4,3] => [1,5,7,6,2,3,4] => ? = 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,6,7,1] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [1,3,5,6,7,2,4] => [1,6,2,7,3,4,5] => ? = 1
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,7,1] => [1,6,7,2,5,4,3] => [1,4,7,6,5,2,3] => ? = 3
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [5,6,2,3,4,7,1] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,3,5,7,1] => [1,6,7,2,4,3,5] => [1,4,6,5,7,2,3] => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,6,7,1] => [1,5,6,7,2,4,3] => [1,5,7,6,2,3,4] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [1,3,4,5,6,7,2] => [1,7,2,3,4,5,6] => ? = 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000470
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 56%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 56%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,4,3,5,2] => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,4,2,5,3] => [1,5,3,4,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,5,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,5,2,4,3] => [1,4,3,5,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,2,4,3] => [1,4,3,5,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,2,5,3] => [1,5,3,4,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [1,4,5,2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,5,3,2,4] => [1,4,5,3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,4,2,5,3] => [1,5,3,4,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,2,3] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,5,4,2,3] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,7,2,6,3,5,4] => [1,5,4,6,3,7,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,6,2,7,3,5,4] => [1,5,4,7,3,6,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [1,6,2,5,3,7,4] => [1,7,4,5,3,6,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,5,3,7,2,6,4] => [1,6,4,7,2,5,3] => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,5,3,4,7,2,6] => [1,6,7,2,4,5,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,2,4,6,3,5] => [1,5,6,3,4,7,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [1,6,2,4,7,3,5] => [1,5,7,3,4,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,4,7,2,5,3,6] => [1,5,3,6,7,2,4] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,7,2,6,3,5,4] => [1,5,4,6,3,7,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,6,2,7,3,5,4] => [1,5,4,7,3,6,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,2,5,4,3,6] => [1,5,4,3,6,7,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [1,6,2,5,4,3,7] => [1,5,4,3,6,2,7] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,5,4,3,7,2,6] => [1,6,7,2,5,4,3] => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [1,6,2,5,7,3,4] => [1,4,7,3,5,6,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,4,3,6,2,5,7] => [1,5,6,2,4,3,7] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,6,2,5,7,3,4] => [1,4,7,3,5,6,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => [1,6,3,4,5,7,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,6,2,4,5,7,3] => [1,7,3,4,5,6,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [1,6,3,4,7,2,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,4,6,2,5,7,3] => [1,7,3,5,6,2,4] => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,4,6,7,2,3,5] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [1,7,2,6,3,5,4] => [1,5,4,6,3,7,2] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,3,5,2,7,4] => [1,7,4,5,2,6,3] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [1,6,3,7,2,5,4] => [1,5,4,7,2,6,3] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [1,5,2,6,3,7,4] => [1,7,4,6,3,5,2] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,6,3,4,5,2,7] => [1,5,2,4,6,3,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [1,6,3,4,7,2,5] => [1,5,7,2,4,6,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => [1,7,2,4,6,3,5] => [1,5,6,3,4,7,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [1,5,2,4,7,3,6] => [1,6,7,3,4,5,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [1,4,7,2,5,3,6] => [1,5,3,6,7,2,4] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [1,7,2,6,4,3,5] => [1,5,6,4,3,7,2] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [1,6,4,3,5,2,7] => [1,5,2,6,4,3,7] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [1,6,4,3,7,2,5] => [1,5,7,2,6,4,3] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [1,5,2,6,4,3,7] => [1,6,4,3,5,2,7] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [1,6,5,2,7,3,4] => [1,4,7,3,6,5,2] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [1,5,2,6,7,3,4] => [1,4,7,3,5,2,6] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [1,4,3,6,7,2,5] => [1,5,7,2,4,3,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [1,6,5,2,7,3,4] => [1,4,7,3,6,5,2] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [1,5,2,6,7,3,4] => [1,4,7,3,5,2,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,6,1] => [1,7,2,4,5,3,6] => [1,5,3,4,6,7,2] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [1,6,7,2,4,5,3] => [1,5,3,4,7,2,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [1,5,2,4,6,7,3] => [1,7,3,4,5,2,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [1,4,6,7,2,5,3] => [1,5,3,7,2,4,6] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,3,4,1] => [1,7,2,6,4,3,5] => [1,5,6,4,3,7,2] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,3,4,1] => [1,6,4,2,7,3,5] => [1,5,7,3,6,4,2] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [6,5,7,2,3,4,1] => [1,6,4,2,5,3,7] => [1,5,3,6,4,2,7] => ? = 3 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000619
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 56%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 56%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0 + 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => [1,4,3,5,2] => 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,4,2,5,3] => [1,5,3,4,2] => 3 = 2 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,3,5,2,4] => [1,4,5,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,5,2,4,3] => [1,4,3,5,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,4,5,2,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,4,5,2,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,2,4,3] => [1,4,3,5,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,4,2,5,3] => [1,5,3,4,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => [1,4,5,2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,5,3,2,4] => [1,4,5,3,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,4,2,5,3] => [1,5,3,4,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,5,4,2,3] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,5,4,2,3] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,3,4,2,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,5,3,2] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => [1,7,2,6,3,5,4] => [1,5,4,6,3,7,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => [1,6,2,7,3,5,4] => [1,5,4,7,3,6,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => [1,6,2,5,3,7,4] => [1,7,4,5,3,6,2] => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => [1,5,3,7,2,6,4] => [1,6,4,7,2,5,3] => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => [1,5,3,4,7,2,6] => [1,6,7,2,4,5,3] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => [1,7,2,4,6,3,5] => [1,5,6,3,4,7,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => [1,6,2,4,7,3,5] => [1,5,7,3,4,6,2] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => [1,4,7,2,5,3,6] => [1,5,3,6,7,2,4] => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => [1,7,2,6,3,5,4] => [1,5,4,6,3,7,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => [1,6,2,7,3,5,4] => [1,5,4,7,3,6,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => [1,7,2,5,4,3,6] => [1,5,4,3,6,7,2] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,2,1] => [1,6,2,5,4,3,7] => [1,5,4,3,6,2,7] => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => [1,5,4,3,7,2,6] => [1,6,7,2,5,4,3] => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => [1,6,2,5,7,3,4] => [1,4,7,3,5,6,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,7,2,1] => [1,4,3,6,2,5,7] => [1,5,6,2,4,3,7] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,4,7,2,1] => [1,6,2,5,7,3,4] => [1,4,7,3,5,6,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => [1,7,2,4,5,6,3] => [1,6,3,4,5,7,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,7,2,1] => [1,6,2,4,5,7,3] => [1,7,3,4,5,6,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,7,2,1] => [1,5,7,2,4,6,3] => [1,6,3,4,7,2,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,7,2,1] => [1,4,6,2,5,7,3] => [1,7,3,5,6,2,4] => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => [1,3,5,7,2,4,6] => [1,4,6,7,2,3,5] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => [1,7,2,6,3,5,4] => [1,5,4,6,3,7,2] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => [1,6,3,5,2,7,4] => [1,7,4,5,2,6,3] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,2,3,1] => [1,6,3,7,2,5,4] => [1,5,4,7,2,6,3] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => [1,5,2,6,3,7,4] => [1,7,4,6,3,5,2] => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,2,3,1] => [1,6,3,4,5,2,7] => [1,5,2,4,6,3,7] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,2,3,1] => [1,6,3,4,7,2,5] => [1,5,7,2,4,6,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,2,3,1] => [1,7,2,4,6,3,5] => [1,5,6,3,4,7,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,2,3,1] => [1,5,2,4,7,3,6] => [1,6,7,3,4,5,2] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => [1,4,7,2,5,3,6] => [1,5,3,6,7,2,4] => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => [1,7,2,6,4,3,5] => [1,5,6,4,3,7,2] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,2,4,1] => [1,6,4,3,5,2,7] => [1,5,2,6,4,3,7] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,2,4,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,2,4,1] => [1,6,4,3,7,2,5] => [1,5,7,2,6,4,3] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,2,4,1] => [1,5,2,6,4,3,7] => [1,6,4,3,5,2,7] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,7,4,3,2,5,1] => [1,6,5,2,7,3,4] => [1,4,7,3,6,5,2] => ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,6,4,3,2,7,1] => [1,5,2,6,7,3,4] => [1,4,7,3,5,2,6] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,5,6,3,2,7,1] => [1,4,3,6,7,2,5] => [1,5,7,2,4,3,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,2,5,1] => [1,6,5,2,7,3,4] => [1,4,7,3,6,5,2] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,7,1] => [1,5,2,6,7,3,4] => [1,4,7,3,5,2,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,2,6,1] => [1,7,2,4,5,3,6] => [1,5,3,4,6,7,2] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => [1,6,7,2,4,5,3] => [1,5,3,4,7,2,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [5,4,3,6,2,7,1] => [1,5,2,4,6,7,3] => [1,7,3,4,5,2,6] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => [1,4,6,7,2,5,3] => [1,5,3,7,2,4,6] => ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,3,4,1] => [1,7,2,6,4,3,5] => [1,5,6,4,3,7,2] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,3,4,1] => [1,6,4,2,7,3,5] => [1,5,7,3,6,4,2] => ? = 3 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => [1,7,2,5,3,6,4] => [1,6,4,5,3,7,2] => ? = 3 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St000711The number of big exceedences of a permutation. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000710The number of big deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001427The number of descents of a signed permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001811The Castelnuovo-Mumford regularity of a permutation. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function.
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