searching the database
Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001085
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0
Description
The number of occurrences of the vincular pattern |21-3 in a permutation.
This is the number of occurrences of the pattern 213, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St000291
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: 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]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1 => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 01 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 01 => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 01 => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 11 => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 011 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 011 => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 011 => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 011 => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 011 => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 101 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 011 => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 011 => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 110 => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 101 => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 011 => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 0111 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 0111 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 0111 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 0111 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 0111 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 0111 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 1011 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 1011 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 0101 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 0101 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 0110 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 0111 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 0111 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 0110 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 0111 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 0111 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 0111 => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,5,4] => 1101 => 1
[]
=> [] => [] => ? => ? = 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] => [2,1,11,10,9,8,7,6,5,4,3] => 1011111111 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [2,1,12,11,10,9,8,7,6,5,4,3] => 10111111111 => ? = 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1 => 0
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 100 => 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 0
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 10000 => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 10000 => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1
[]
=> []
=> [] => ? => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 0
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,8,1] => 1100000001 => ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => ? = 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,1,1,1] => 1000000111 => ? = 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,1,1,1] => 1000000111 => ? = 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,1,1,1,1] => 1000001111 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1,1] => 1000111111 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,8] => 1010000000 => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [11,1] => 100000000001 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1,1,1] => ? => ? = 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1 => 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 10 => 1 = 0 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 11 => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 100 => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 100 => 1 = 0 + 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2,1] => 101 => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 110 => 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 111 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1000 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1001 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1001 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 1010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 1100 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 1100 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1011 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1101 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1110 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1111 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5] => 10000 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 10001 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => 10010 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => 10010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 10100 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 11000 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 11000 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => 10100 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 11000 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 11000 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 11000 => 1 = 0 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1] => 1001111111 => ? = 1 + 1
[]
=> []
=> [] => ? => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => 1100000000 => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,8,1] => 1100000001 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10] => 11000000000 => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [9,1] => 1000000001 => ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,1,1] => 1000000011 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [7,1,1,1] => 1000000111 => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? => ? => ? = 1 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,1,1,1] => 1000000111 => ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [6,1,1,1,1] => 1000001111 => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [5,1,1,1,1,1] => 1000011111 => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [4,1,1,1,1,1,1] => 1000111111 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,8] => 1010000000 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => 10000000001 => ? = 1 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> [11,1] => 100000000001 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [3,1,1,1,1,1,1,1,1,1] => ? => ? = 1 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000201
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [.,.]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => [2,8,7,6,5,1,4,3] => [[.,[[[[.,.],.],.],.]],[[.,.],.]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,6,1,8,7] => [2,8,7,6,5,1,4,3] => [[.,[[[[.,.],.],.],.]],[[.,.],.]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,1,8] => [2,8,7,6,5,4,1,3] => [[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 0 + 1
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,5,6,8,7,4,1] => [2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 0 + 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,6,8,7,5,4,3,1] => [2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 0 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,5,8,7,6] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [3,2,1,6,5,8,7,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,2,1,8,7,6,5,4] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5,8,7] => [3,2,8,1,7,6,5,4] => [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,2,4,1,6,8,7,5] => [3,2,8,1,7,6,5,4] => [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,2,4,1,7,6,8,5] => [3,2,8,1,7,6,5,4] => [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> ? = 2 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,1] => [3,2,8,7,6,5,4,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,5,4,1,8,7,6] => [3,2,8,7,1,6,5,4] => [[[.,.],[[.,.],.]],[[[.,.],.],.]]
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,5,4,6,1,8,7] => [3,2,8,7,6,1,5,4] => [[[.,.],[[[.,.],.],.]],[[.,.],.]]
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,5,4,7,6,1,8] => [3,2,8,7,6,5,1,4] => [[[.,.],[[[[.,.],.],.],.]],[.,.]]
=> ? = 2 + 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [3,2,5,4,7,6,8,1] => [3,2,8,7,6,5,4,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,2,6,5,4,8,7,1] => [3,2,8,7,6,5,4,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> [3,2,6,5,7,4,8,1] => [3,2,8,7,6,5,4,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 1 + 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [3,2,7,6,5,4,1,8] => [3,2,8,7,6,5,1,4] => [[[.,.],[[[[.,.],.],.],.]],[.,.]]
=> ? = 2 + 1
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [3,4,2,1,7,8,6,5] => [3,8,2,1,7,6,5,4] => [[[.,[.,.]],.],[[[[.,.],.],.],.]]
=> ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,1] => [3,8,2,7,6,5,4,1] => [[[.,[.,.]],[[[[.,.],.],.],.]],.]
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [3,4,5,2,6,7,8,1] => [3,8,7,2,6,5,4,1] => [[[.,[[.,.],.]],[[[.,.],.],.]],.]
=> ? = 1 + 1
[1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,5,8,7,6,4,2,1] => [3,8,7,6,5,4,2,1] => [[[.,[[[[[.,.],.],.],.],.]],.],.]
=> ? = 0 + 1
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
=> [4,3,6,5,7,2,8,1] => [4,3,8,7,6,2,5,1] => [[[[.,.],[[[.,.],.],.]],[.,.]],.]
=> ? = 2 + 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [4,3,6,5,8,7,2,1] => [4,3,8,7,6,5,2,1] => [[[[.,.],[[[[.,.],.],.],.]],.],.]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => [[[[[.,.],.],.],.],[[[.,.],.],.]]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
=> [5,4,3,2,7,6,1,8] => [5,4,3,2,8,7,1,6] => [[[[[.,.],.],.],[[.,.],.]],[.,.]]
=> ? = 2 + 1
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [5,4,6,3,7,2,8,1] => [5,4,8,3,7,2,6,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> ? = 3 + 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [6,5,7,4,3,2,8,1] => [6,5,8,4,3,2,7,1] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [6,5,7,4,8,3,2,1] => [6,5,8,4,7,3,2,1] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [6,5,8,7,4,3,2,1] => [6,5,8,7,4,3,2,1] => [[[[[[.,.],[[.,.],.]],.],.],.],.]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,8,1] => [[[[[[[.,.],.],.],.],.],[.,.]],.]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [7,6,5,4,8,3,2,1] => [7,6,5,4,8,3,2,1] => [[[[[[[.,.],.],.],[.,.]],.],.],.]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [7,6,8,5,4,3,2,1] => [7,6,8,5,4,3,2,1] => [[[[[[[.,.],[.,.]],.],.],.],.],.]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,9,5,4,3,2,1,8] => ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,7,5,4,3,2,1,9] => [6,9,8,5,4,3,2,1,7] => ?
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,9,8] => [[[[[[[.,.],.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => [[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,10,9,8,7,6,5,4,3] => [2,1,10,9,8,7,6,5,4,3] => [[.,.],[[[[[[[[.,.],.],.],.],.],.],.],.]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1,9,10] => [8,7,6,5,4,3,2,1,10,9] => [[[[[[[[.,.],.],.],.],.],.],.],[[.,.],.]]
=> ? = 1 + 1
[]
=> [] => [] => .
=> ? = 0 + 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] => [2,9,8,7,6,5,4,3,1] => [[.,[[[[[[[.,.],.],.],.],.],.],.]],.]
=> ? = 0 + 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] => [3,2,9,8,7,6,5,4,1] => [[[.,.],[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,9,2,8,7,6,5,4,1] => [[[.,[.,.]],[[[[[.,.],.],.],.],.]],.]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [2,10,9,8,7,6,5,4,3,1] => [[.,[[[[[[[[.,.],.],.],.],.],.],.],.]],.]
=> ? = 0 + 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,10,9,8,7,6,5,4,1] => [[[.,.],[[[[[[[.,.],.],.],.],.],.],.]],.]
=> ? = 1 + 1
Description
The number of leaf nodes in a binary tree.
Equivalently, the number of cherries [1] in the complete binary tree.
The number of binary trees of size n, at least 1, with exactly one leaf node for is 2n−1, see [2].
The number of binary tree of size n, at least 3, with exactly two leaf nodes is n(n+1)2n−2, see [3].
Matching statistic: St000196
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => [.,.]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [.,[.,.]]
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => [[.,.],.]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [[[.,.],[.,.]],.]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [[[.,[.,.]],.],.]
=> 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> 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] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,2,6,7,8,5,4,3] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,5,7,6,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,5,4,6,7,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,6,7,5,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,5,7,8,6,4,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,7,6,5,8] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,4,3,6,5,8,7,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,4,3,8,7,6,5,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,5,8,7,6,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,6,5,4,3,8,7,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,6,7,8,5,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,7,5,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,7,6,5,8,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,6,8,5,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => [.,[[[[[[[.,.],.],.],.],.],.],.]]
=> ? = 0
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7,8] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,6,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,3,6,8,7,5] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,4,3,7,6,8,5] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,8,7,6,5] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,1,4,6,5,3,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [2,1,5,4,6,3,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [2,1,5,4,7,6,8,3] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,6,5,4,3,8,7] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [2,1,6,5,7,4,8,3] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,8,7,6,5,4,3] => [2,1,8,7,6,5,4,3] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,1,4,5,7,8,6] => [2,8,1,7,6,5,4,3] => [[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> ? = 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [2,3,4,1,6,7,8,5] => [2,8,7,1,6,5,4,3] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,4,5,1,6,7,8] => [2,8,7,6,1,5,4,3] => [[.,[[[.,.],.],.]],[[[.,.],.],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,1,7,8] => [2,8,7,6,5,1,4,3] => [[.,[[[[.,.],.],.],.]],[[.,.],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [2,3,4,5,6,1,8,7] => [2,8,7,6,5,1,4,3] => [[.,[[[[.,.],.],.],.]],[[.,.],.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,1,8] => [2,8,7,6,5,4,1,3] => [[.,[[[[[.,.],.],.],.],.]],[.,.]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => [2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 0
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [2,3,5,6,8,7,4,1] => [2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 0
[1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [2,4,3,1,6,5,8,7] => [2,8,7,1,6,5,4,3] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [2,4,3,1,6,8,7,5] => [2,8,7,1,6,5,4,3] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> ? = 1
[1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,3,1,7,6,8,5] => [2,8,7,1,6,5,4,3] => [[.,[[.,.],.]],[[[[.,.],.],.],.]]
=> ? = 1
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,6,8,7,5,4,3,1] => [2,8,7,6,5,4,3,1] => [[.,[[[[[[.,.],.],.],.],.],.]],.]
=> ? = 0
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6,7,8] => [3,2,1,8,7,6,5,4] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by 2n−2.
Matching statistic: St000353
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 75%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000353: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [2,3,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [2,3,1] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [1,3,2] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [3,4,5,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [3,4,5,1,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [3,4,5,1,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [3,4,1,5,2] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [3,4,1,5,2] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [1,3,4,5,2] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [3,1,4,5,2] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [1,3,4,5,2] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [1,3,4,5,2] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [1,3,4,5,2] => 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,5,4] => [4,5,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
Description
The number of inner valleys of a permutation.
The number of valleys including the boundary is [[St000099]].
Matching statistic: St000023
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [3,1,2] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 0
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1,3] => 0
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,3,1,2] => 0
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [2,4,1,3] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [4,2,1,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [4,2,1,3] => 0
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,4,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,2,1,4] => 0
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,1,2] => 0
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [5,2,1,4,3] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [5,2,1,4,3] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [5,2,1,4,3] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [5,2,1,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [5,2,1,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [5,2,4,1,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [5,2,4,1,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [2,5,4,1,3] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [5,4,2,1,3] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [5,4,2,1,3] => 0
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [2,5,4,1,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [5,4,2,1,3] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => [5,4,2,1,3] => 0
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [5,4,2,1,3] => 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 0
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000092
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1 = 0 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 1 = 0 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,4,3,1] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,5,4] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,4,3,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,4,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,6,4,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,5,4,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,7,6,5,4,3,2] => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,7,6,5,4,3,2] => ? = 0 + 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation w=[w1,...,wn] is either a position i such that wi−1<wi>wi+1 or 1 if w1>w2 or n if wn>wn−1.
In other words, it is a peak in the word [0,w1,...,wn,0].
Matching statistic: St001181
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001181: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 75%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
[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]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 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]
=> [1,1,1,0,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 0
Description
Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000099The number of valleys of a permutation, including the boundary. St000837The number of ascents of distance 2 of a permutation. St000022The number of fixed points of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001487The number of inner corners of a skew partition. St000664The number of right ropes of a permutation. St001964The interval resolution global dimension of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001208The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(xn). St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!