searching the database
Your data matches 46 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: St000035
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 0
[2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => 0
[2,1,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => 0
[3,1,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => 1
[3,2,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => 1
[3,4,1,2] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => 1
[4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => 1
[4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => 1
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000292
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => ? = 0
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,2,3] => 00 => 0
[2,1,3] => [1,2,3] => 00 => 0
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,3,2] => 01 => 1
[3,2,1] => [1,3,2] => 01 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [1,2,3,4] => 000 => 0
[1,4,2,3] => [1,2,4,3] => 001 => 1
[1,4,3,2] => [1,2,4,3] => 001 => 1
[2,1,3,4] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [1,2,3,4] => 000 => 0
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,2,4,3] => 001 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,3,2,4] => 010 => 1
[3,1,4,2] => [1,3,4,2] => 001 => 1
[3,2,1,4] => [1,3,2,4] => 010 => 1
[3,2,4,1] => [1,3,4,2] => 001 => 1
[3,4,1,2] => [1,3,2,4] => 010 => 1
[3,4,2,1] => [1,3,2,4] => 010 => 1
[4,1,2,3] => [1,4,3,2] => 011 => 1
[4,1,3,2] => [1,4,2,3] => 010 => 1
[4,2,1,3] => [1,4,3,2] => 011 => 1
[4,2,3,1] => [1,4,2,3] => 010 => 1
[4,3,1,2] => [1,4,2,3] => 010 => 1
[4,3,2,1] => [1,4,2,3] => 010 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 1
[] => [] => ? => ? = 0
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => ? = 0
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,2,3] => 00 => 0
[2,1,3] => [1,2,3] => 00 => 0
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,3,2] => 01 => 1
[3,2,1] => [1,3,2] => 01 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,3,4] => 000 => 0
[1,3,2,4] => [1,2,3,4] => 000 => 0
[1,3,4,2] => [1,2,3,4] => 000 => 0
[1,4,2,3] => [1,2,4,3] => 001 => 1
[1,4,3,2] => [1,2,4,3] => 001 => 1
[2,1,3,4] => [1,2,3,4] => 000 => 0
[2,1,4,3] => [1,2,3,4] => 000 => 0
[2,3,1,4] => [1,2,3,4] => 000 => 0
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,2,4,3] => 001 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,3,2,4] => 010 => 1
[3,1,4,2] => [1,3,4,2] => 001 => 1
[3,2,1,4] => [1,3,2,4] => 010 => 1
[3,2,4,1] => [1,3,4,2] => 001 => 1
[3,4,1,2] => [1,3,2,4] => 010 => 1
[3,4,2,1] => [1,3,2,4] => 010 => 1
[4,1,2,3] => [1,4,3,2] => 011 => 1
[4,1,3,2] => [1,4,2,3] => 010 => 1
[4,2,1,3] => [1,4,3,2] => 011 => 1
[4,2,3,1] => [1,4,2,3] => 010 => 1
[4,3,1,2] => [1,4,2,3] => 010 => 1
[4,3,2,1] => [1,4,2,3] => 010 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 1
[] => [] => ? => ? = 0
Description
The number of runs of ones in a binary word.
Matching statistic: St000291
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => => => ? = 0
[1,2] => [1,2] => 0 => 0 => 0
[2,1] => [1,2] => 0 => 0 => 0
[1,2,3] => [1,2,3] => 00 => 00 => 0
[1,3,2] => [1,2,3] => 00 => 00 => 0
[2,1,3] => [1,2,3] => 00 => 00 => 0
[2,3,1] => [1,2,3] => 00 => 00 => 0
[3,1,2] => [1,3,2] => 01 => 10 => 1
[3,2,1] => [1,3,2] => 01 => 10 => 1
[1,2,3,4] => [1,2,3,4] => 000 => 000 => 0
[1,2,4,3] => [1,2,3,4] => 000 => 000 => 0
[1,3,2,4] => [1,2,3,4] => 000 => 000 => 0
[1,3,4,2] => [1,2,3,4] => 000 => 000 => 0
[1,4,2,3] => [1,2,4,3] => 001 => 100 => 1
[1,4,3,2] => [1,2,4,3] => 001 => 100 => 1
[2,1,3,4] => [1,2,3,4] => 000 => 000 => 0
[2,1,4,3] => [1,2,3,4] => 000 => 000 => 0
[2,3,1,4] => [1,2,3,4] => 000 => 000 => 0
[2,3,4,1] => [1,2,3,4] => 000 => 000 => 0
[2,4,1,3] => [1,2,4,3] => 001 => 100 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 100 => 1
[3,1,2,4] => [1,3,2,4] => 010 => 010 => 1
[3,1,4,2] => [1,3,4,2] => 001 => 100 => 1
[3,2,1,4] => [1,3,2,4] => 010 => 010 => 1
[3,2,4,1] => [1,3,4,2] => 001 => 100 => 1
[3,4,1,2] => [1,3,2,4] => 010 => 010 => 1
[3,4,2,1] => [1,3,2,4] => 010 => 010 => 1
[4,1,2,3] => [1,4,3,2] => 011 => 110 => 1
[4,1,3,2] => [1,4,2,3] => 010 => 010 => 1
[4,2,1,3] => [1,4,3,2] => 011 => 110 => 1
[4,2,3,1] => [1,4,2,3] => 010 => 010 => 1
[4,3,1,2] => [1,4,2,3] => 010 => 010 => 1
[4,3,2,1] => [1,4,2,3] => 010 => 010 => 1
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,3,5,4] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,4,3,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,4,5,3] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,2,5,3,4] => [1,2,3,5,4] => 0001 => 1000 => 1
[1,2,5,4,3] => [1,2,3,5,4] => 0001 => 1000 => 1
[1,3,2,4,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,2,5,4] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,4,2,5] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,4,5,2] => [1,2,3,4,5] => 0000 => 0000 => 0
[1,3,5,2,4] => [1,2,3,5,4] => 0001 => 1000 => 1
[1,3,5,4,2] => [1,2,3,5,4] => 0001 => 1000 => 1
[1,4,2,3,5] => [1,2,4,3,5] => 0010 => 0100 => 1
[1,4,2,5,3] => [1,2,4,5,3] => 0001 => 1000 => 1
[1,4,3,2,5] => [1,2,4,3,5] => 0010 => 0100 => 1
[1,4,3,5,2] => [1,2,4,5,3] => 0001 => 1000 => 1
[1,4,5,2,3] => [1,2,4,3,5] => 0010 => 0100 => 1
[1,4,5,3,2] => [1,2,4,3,5] => 0010 => 0100 => 1
[] => [] => ? => ? => ? = 0
Description
The number of descents of a binary word.
Matching statistic: St000386
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 0
[1,2] => [1,2] => [2] => [1,1,0,0]
=> 0
[2,1] => [1,2] => [2] => [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,1,3] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[2,3,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,3,2,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,3,4,2] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,4,2,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,4,3,2] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,3,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,1,4,3] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,1,4] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,3,4,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,1,4] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,2,4,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,1,2,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,2,1,3] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[4,2,3,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,3,1,2] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[4,3,2,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,3,4,5,6,2,8,1] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,3,2,4,5,6,8,1] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,2,3,4,5,6,8,1] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,3,4,5,7,8,1,2] => [1,6,8,2,3,4,5,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,3,4,5,2,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,3,2,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,2,3,4,5,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[4,3,2,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[4,2,3,5,6,7,1,8] => [1,4,5,6,7,2,3,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
[5,2,3,4,8,6,7,1] => [1,5,8,2,3,4,6,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[5,2,3,4,8,7,6,1] => [1,5,8,2,3,4,6,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,2,3,4,5,8,7,1] => [1,6,8,2,3,4,5,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,2,3,5,4,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[5,2,4,3,8,7,6,1] => [1,5,8,2,3,4,6,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,2,4,3,6,5,8,1] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,2,4,5,3,8,7,1] => [1,6,8,2,3,4,5,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,2,4,5,3,6,8,1] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,3,2,5,4,7,1,8] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,3,2,5,4,8,7,1] => [1,6,8,2,3,4,5,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,3,2,5,6,4,8,1] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[5,3,4,2,8,6,7,1] => [1,5,8,2,3,4,6,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[7,3,4,2,6,5,8,1] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[6,3,4,5,2,8,7,1] => [1,6,8,2,3,4,5,7] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[2,4,6,1,3,5,8,7] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[2,4,6,1,3,5,7,8] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[2,4,6,1,7,3,8,5] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[2,6,1,3,4,7,8,5] => [1,2,6,7,8,5,4,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[3,1,5,2,7,4,8,6] => [1,3,5,7,8,6,4,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,4,6,2,7,3,5,8] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,4,6,2,7,3,8,5] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,4,6,2,3,5,8,7] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[1,4,6,2,3,5,7,8] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[6,1,7,2,3,8,4,5] => [1,6,8,5,3,7,4,2] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 2
[6,1,2,3,4,7,8,5] => [1,6,7,8,5,4,3,2] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,6,2,3,4,7,8,5] => [1,2,6,7,8,5,4,3] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[7,1,4,2,3,5,8,6] => [1,7,8,6,5,3,4,2] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 2
[6,1,4,5,2,7,8,3] => [1,6,7,8,3,4,5,2] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 2
[2,8,1,5,6,3,4,7] => [1,2,8,7,4,5,6,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[4,3,1,8,2,7,5,6] => [1,4,8,6,7,5,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[2,8,4,5,1,3,6,7] => [1,2,8,7,6,3,4,5] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[7,3,4,5,6,1,8,2] => [1,7,8,2,3,4,5,6] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[9,1,2,3,6,4,5,7,8] => [1,9,8,7,5,6,4,3,2] => [2,1,1,2,1,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[8,3,4,5,6,7,1,9,2] => [1,8,9,2,3,4,5,6,7] => [3,6] => [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[] => [] => [] => ?
=> ? = 0
[5,1,7,2,8,4,6,3] => [1,5,8,3,7,6,4,2] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 2
[3,2,5,1,7,4,8,6] => [1,3,5,7,8,6,4,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 1
[3,6,4,1,8,7,5,2] => [1,3,4,2,6,7,5,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 2
[2,4,6,3,7,5,8,1] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2
[6,3,4,5,8,7,1,2] => [1,6,7,2,3,4,5,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000994
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,2,3] => [1,2,3] => 0
[2,1,3] => [1,2,3] => [1,2,3] => 0
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,3,2] => [1,3,2] => 1
[3,2,1] => [1,3,2] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,1,2,3] => [1,4,3,2] => [1,3,4,2] => 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,2,1,3] => [1,4,3,2] => [1,3,4,2] => 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => 1
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[3,1,7,4,5,6,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,4,6,5,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,5,4,6,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,1,7,5,6,4,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,4,5,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,4,6,5,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,5,4,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,2,7,5,6,4,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,1,5,6,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,1,6,5,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,2,5,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,2,6,5,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,1,6,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,2,6,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,6,1,2] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[3,4,7,5,6,2,1] => [1,3,7,2,4,5,6] => [1,7,3,2,4,5,6] => ? = 1
[4,1,3,6,5,7,2] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,1,3,7,5,6,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,1,3,7,6,5,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,1,5,6,3,7,2] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,1,5,7,3,6,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,1,5,7,6,3,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,2,3,6,5,7,1] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,2,3,7,5,6,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,2,3,7,6,5,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,2,5,6,3,7,1] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,2,5,7,3,6,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,2,5,7,6,3,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,1,6,5,7,2] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,3,1,7,5,6,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,1,7,6,5,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,2,6,5,7,1] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,3,2,7,5,6,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,2,7,6,5,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,5,6,1,7,2] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,3,5,6,2,7,1] => [1,4,6,7,2,3,5] => [1,7,2,4,3,6,5] => ? = 1
[4,3,5,7,1,6,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,5,7,2,6,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,5,7,6,1,2] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[4,3,5,7,6,2,1] => [1,4,7,2,3,5,6] => [1,7,2,4,3,5,6] => ? = 1
[5,1,3,4,6,7,2] => [1,5,6,7,2,3,4] => [1,7,2,3,5,6,4] => ? = 1
[5,1,3,4,7,6,2] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[5,1,3,6,7,4,2] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[5,1,4,3,6,7,2] => [1,5,6,7,2,3,4] => [1,7,2,3,5,6,4] => ? = 1
[5,1,4,3,7,6,2] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[5,1,4,6,7,3,2] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[5,2,3,4,6,7,1] => [1,5,6,7,2,3,4] => [1,7,2,3,5,6,4] => ? = 1
[5,2,3,4,7,6,1] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[5,2,3,6,7,4,1] => [1,5,7,2,3,4,6] => [1,7,2,3,5,4,6] => ? = 1
[5,2,4,3,6,7,1] => [1,5,6,7,2,3,4] => [1,7,2,3,5,6,4] => ? = 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000201
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 55% ●values known / values provided: 55%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [.,.]
=> 1 = 0 + 1
[1,2] => [1,2] => [2,1] => [[.,.],.]
=> 1 = 0 + 1
[2,1] => [1,2] => [2,1] => [[.,.],.]
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[1,3,2] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[2,1,3] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 1 = 0 + 1
[3,1,2] => [1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 2 = 1 + 1
[3,2,1] => [1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 1 = 0 + 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 2 = 1 + 1
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [8,2,7,1,6,3,5,4] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3 + 1
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3 + 1
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [8,2,7,3,6,1,5,4] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> ? = 3 + 1
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3 + 1
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3 + 1
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3 + 1
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [8,2,7,3,6,4,5,1] => [[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> ? = 3 + 1
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [8,2,7,1,6,3,5,4] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3 + 1
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3 + 1
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [8,2,7,3,6,1,5,4] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> ? = 3 + 1
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2 + 1
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [8,2,7,3,6,5,4,1] => [[[.,.],[[.,.],[[[.,.],.],.]]],.]
=> ? = 2 + 1
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [8,2,7,1,6,3,5,4] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3 + 1
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3 + 1
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3 + 1
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3 + 1
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3 + 1
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3 + 1
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3 + 1
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => [8,2,7,1,6,5,4,3] => [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> ? = 2 + 1
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => [8,1,7,4,6,5,3,2] => [[.,.],[[[[.,.],[[.,.],.]],.],.]]
=> ? = 2 + 1
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => [8,2,7,4,6,5,3,1] => [[[.,.],[[[.,.],[[.,.],.]],.]],.]
=> ? = 2 + 1
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => [8,2,7,6,5,4,3,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 1 + 1
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => [8,3,1,7,6,5,4,2] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1 + 1
[8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3 + 1
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3 + 1
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3 + 1
[8,6,7,4,5,2,3,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3 + 1
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2 + 1
[8,6,7,5,3,2,4,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3 + 1
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3 + 1
[8,6,7,5,2,3,4,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3 + 1
[8,6,5,7,2,3,4,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3 + 1
[5,6,7,8,2,3,4,1] => [1,5,2,6,3,7,4,8] => [8,4,7,3,6,2,5,1] => [[[[[.,.],[.,.]],[.,.]],[.,.]],.]
=> ? = 3 + 1
[8,6,5,4,3,2,7,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3 + 1
[8,6,4,5,3,2,7,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2 + 1
[8,5,4,6,3,2,7,1] => [1,8,2,5,3,4,6,7] => [8,1,7,4,6,5,3,2] => [[.,.],[[[[.,.],[[.,.],.]],.],.]]
=> ? = 2 + 1
[8,6,5,3,4,2,7,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3 + 1
[8,6,4,3,5,2,7,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2 + 1
[8,6,3,4,5,2,7,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2 + 1
[8,4,5,3,6,2,7,1] => [1,8,2,4,3,5,6,7] => [8,1,7,5,6,4,3,2] => [[.,.],[[[[[.,.],[.,.]],.],.],.]]
=> ? = 2 + 1
[8,6,5,4,2,3,7,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3 + 1
[8,6,4,5,2,3,7,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2 + 1
[8,5,4,3,2,6,7,1] => [1,8,2,5,3,4,6,7] => [8,1,7,4,6,5,3,2] => [[.,.],[[[[.,.],[[.,.],.]],.],.]]
=> ? = 2 + 1
[8,4,5,3,2,6,7,1] => [1,8,2,4,3,5,6,7] => [8,1,7,5,6,4,3,2] => [[.,.],[[[[[.,.],[.,.]],.],.],.]]
=> ? = 2 + 1
[8,5,3,4,2,6,7,1] => [1,8,2,5,3,4,6,7] => [8,1,7,4,6,5,3,2] => [[.,.],[[[[.,.],[[.,.],.]],.],.]]
=> ? = 2 + 1
[8,5,4,2,3,6,7,1] => [1,8,2,5,3,4,6,7] => [8,1,7,4,6,5,3,2] => [[.,.],[[[[.,.],[[.,.],.]],.],.]]
=> ? = 2 + 1
[8,4,5,2,3,6,7,1] => [1,8,2,4,3,5,6,7] => [8,1,7,5,6,4,3,2] => [[.,.],[[[[[.,.],[.,.]],.],.],.]]
=> ? = 2 + 1
[8,4,3,2,5,6,7,1] => [1,8,2,4,3,5,6,7] => [8,1,7,5,6,4,3,2] => [[.,.],[[[[[.,.],[.,.]],.],.],.]]
=> ? = 2 + 1
[8,4,2,3,5,6,7,1] => [1,8,2,4,3,5,6,7] => [8,1,7,5,6,4,3,2] => [[.,.],[[[[[.,.],[.,.]],.],.],.]]
=> ? = 2 + 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 $2^{n-1}$, see [2].
The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
Matching statistic: St000834
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[2,1,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[2,3,1] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[3,1,2] => [1,3,2] => [2,3,1] => [2,1,3] => 1
[3,2,1] => [1,3,2] => [2,3,1] => [2,1,3] => 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [3,2,1,4] => 1
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [3,2,1,4] => 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [3,2,1,4] => 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [3,2,1,4] => 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [4,2,1,3] => 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [4,2,1,3] => 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [2,1,3,4] => 1
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [2,1,3,4] => 1
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 1
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 1
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [3,2,4,1] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1,5] => 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1,5] => 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1,5] => 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1,5] => 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,2,4,1] => 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [5,3,2,1,4] => 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,2,4,1] => 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [5,3,2,1,4] => 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,2,4,1] => 1
[1,2,3,6,4,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,3,6,5,4,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,3,6,7,4,5] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,3,6,7,5,4] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,4,6,3,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,4,6,5,3,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,4,6,7,5,3] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,2,6,7,3,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,2,6,7,4,3,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,2,6,7,5,3,4] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,2,6,7,5,4,3] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,3,2,6,4,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,2,6,5,4,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,2,6,7,4,5] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,2,6,7,5,4] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,4,6,2,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,4,6,5,2,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,4,6,7,2,5] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,4,6,7,5,2] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[1,3,6,7,2,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,3,6,7,4,2,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,3,6,7,5,2,4] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,3,6,7,5,4,2] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[1,6,2,7,4,3,5] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[1,6,2,7,5,3,4] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[1,6,3,7,4,2,5] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[1,6,3,7,5,2,4] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[1,6,4,7,2,3,5] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[1,6,4,7,3,2,5] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[1,6,4,7,5,2,3] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[1,6,4,7,5,3,2] => [1,2,6,3,4,7,5] => [5,7,4,3,6,2,1] => [5,4,3,7,2,1,6] => ? = 2
[2,1,3,6,4,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,3,6,5,4,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,3,6,7,4,5] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,3,6,7,5,4] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,4,6,3,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,4,6,5,3,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,4,6,7,3,5] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,4,6,7,5,3] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,1,6,7,3,4,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[2,1,6,7,4,3,5] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[2,1,6,7,5,3,4] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[2,1,6,7,5,4,3] => [1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [5,4,3,2,7,1,6] => ? = 2
[2,3,1,6,4,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,3,1,6,5,4,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,3,1,6,7,4,5] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,3,1,6,7,5,4] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,3,4,6,1,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
[2,3,4,6,5,1,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [7,5,4,3,2,6,1] => ? = 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St000251
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000251: Set partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> ? = 0
[1,2] => [1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [1,2] => [1,2] => {{1},{2}}
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[3,1,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[3,2,1] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [1,7,6,5,8,4,3,2] => {{1},{2,7},{3,6},{4,5,8}}
=> ? = 3
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [1,7,6,5,8,4,2,3] => ?
=> ? = 3
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [1,6,7,5,8,3,4,2] => ?
=> ? = 3
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [1,6,7,5,8,3,2,4] => ?
=> ? = 3
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [1,7,5,6,4,8,3,2] => ?
=> ? = 3
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [1,7,5,6,4,8,2,3] => ?
=> ? = 3
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [1,6,5,7,4,3,8,2] => ?
=> ? = 3
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [1,6,5,7,4,3,2,8] => ?
=> ? = 3
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [1,7,6,5,8,4,3,2] => {{1},{2,7},{3,6},{4,5,8}}
=> ? = 3
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [1,7,6,5,8,4,2,3] => ?
=> ? = 3
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [1,6,7,5,8,3,4,2] => ?
=> ? = 3
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [1,6,7,5,8,3,2,4] => ?
=> ? = 3
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [1,7,4,5,6,8,3,2] => ?
=> ? = 2
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [1,6,4,5,7,3,8,2] => ?
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [1,6,4,5,7,3,2,8] => ?
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [1,7,6,5,8,4,3,2] => {{1},{2,7},{3,6},{4,5,8}}
=> ? = 3
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [1,7,6,5,8,4,2,3] => ?
=> ? = 3
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [1,7,5,6,4,8,3,2] => ?
=> ? = 3
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [1,7,5,6,4,8,2,3] => ?
=> ? = 3
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [1,5,6,7,3,4,8,2] => ?
=> ? = 3
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [1,6,5,7,3,4,2,8] => {{1},{2,6},{3,5},{4,7},{8}}
=> ? = 3
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [1,7,5,6,4,8,3,2] => ?
=> ? = 3
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [1,7,5,6,4,8,2,3] => ?
=> ? = 3
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [1,7,4,5,6,8,3,2] => ?
=> ? = 2
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [1,7,5,6,4,8,3,2] => ?
=> ? = 3
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [1,7,5,6,4,8,2,3] => ?
=> ? = 3
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [1,7,4,5,6,8,3,2] => ?
=> ? = 2
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => [1,7,4,5,6,8,3,2] => ?
=> ? = 2
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => [1,7,4,5,6,8,2,3] => ?
=> ? = 2
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => [1,5,4,6,3,7,8,2] => ?
=> ? = 2
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => [1,5,4,6,3,7,2,8] => ?
=> ? = 2
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => [1,3,4,5,6,7,2,8] => ?
=> ? = 1
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => [1,3,4,5,7,6,8,2] => ?
=> ? = 1
[8,7,6,5,4,2,3,1] => [1,8,2,7,3,6,4,5] => [1,7,6,5,8,4,3,2] => {{1},{2,7},{3,6},{4,5,8}}
=> ? = 3
[8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => [1,6,7,5,8,3,4,2] => ?
=> ? = 3
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => [1,7,5,6,4,8,3,2] => ?
=> ? = 3
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => [1,6,5,7,4,3,8,2] => ?
=> ? = 3
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => [1,6,5,7,4,2,3,8] => ?
=> ? = 3
[8,7,6,4,5,2,3,1] => [1,8,2,7,3,6,4,5] => [1,7,6,5,8,4,3,2] => {{1},{2,7},{3,6},{4,5,8}}
=> ? = 3
[8,6,7,4,5,2,3,1] => [1,8,2,6,3,7,4,5] => [1,6,7,5,8,3,4,2] => ?
=> ? = 3
[6,7,8,4,5,2,3,1] => [1,6,2,7,3,8,4,5] => [1,6,7,5,8,2,3,4] => ?
=> ? = 3
[8,7,4,5,6,2,3,1] => [1,8,2,7,3,4,5,6] => [1,7,4,5,6,8,3,2] => ?
=> ? = 2
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => [1,6,4,5,7,3,8,2] => ?
=> ? = 2
[6,7,4,5,8,2,3,1] => [1,6,2,7,3,4,5,8] => [1,6,4,5,7,2,3,8] => ?
=> ? = 2
[8,6,7,5,3,2,4,1] => [1,8,2,6,3,7,4,5] => [1,6,7,5,8,3,4,2] => ?
=> ? = 3
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [1,6,5,7,4,3,8,2] => ?
=> ? = 3
[8,5,6,7,3,2,4,1] => [1,8,2,5,3,6,4,7] => [1,5,6,7,3,4,8,2] => ?
=> ? = 3
[8,6,7,5,2,3,4,1] => [1,8,2,6,3,7,4,5] => [1,6,7,5,8,3,4,2] => ?
=> ? = 3
[8,6,5,7,2,3,4,1] => [1,8,2,6,3,5,4,7] => [1,6,5,7,4,3,8,2] => ?
=> ? = 3
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St000196
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [.,.]
=> 0
[1,2] => [1,2] => [2,1] => [[.,.],.]
=> 0
[2,1] => [1,2] => [2,1] => [[.,.],.]
=> 0
[1,2,3] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[1,3,2] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[2,1,3] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[2,3,1] => [1,2,3] => [3,2,1] => [[[.,.],.],.]
=> 0
[3,1,2] => [1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[3,2,1] => [1,3,2] => [3,1,2] => [[.,.],[.,.]]
=> 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 0
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [[[.,.],.],[.,.]]
=> 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [[[.,.],.],[.,.]]
=> 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [[.,.],[.,[.,.]]]
=> 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [[.,.],[[.,.],.]]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> 1
[8,7,6,5,4,3,2,1] => [1,8,2,7,3,6,4,5] => [8,1,7,2,6,3,5,4] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[7,8,6,5,4,3,2,1] => [1,7,2,8,3,6,4,5] => [8,2,7,1,6,3,5,4] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,6,7,5,4,3,2,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3
[7,6,8,5,4,3,2,1] => [1,7,2,6,3,8,4,5] => [8,2,7,3,6,1,5,4] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> ? = 3
[8,7,5,6,4,3,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3
[7,8,5,6,4,3,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3
[8,6,5,7,4,3,2,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3
[7,6,5,8,4,3,2,1] => [1,7,2,6,3,5,4,8] => [8,2,7,3,6,4,5,1] => [[[.,.],[[.,.],[[.,.],[.,.]]]],.]
=> ? = 3
[8,7,6,4,5,3,2,1] => [1,8,2,7,3,6,4,5] => [8,1,7,2,6,3,5,4] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[7,8,6,4,5,3,2,1] => [1,7,2,8,3,6,4,5] => [8,2,7,1,6,3,5,4] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,6,7,4,5,3,2,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3
[7,6,8,4,5,3,2,1] => [1,7,2,6,3,8,4,5] => [8,2,7,3,6,1,5,4] => [[[.,.],[[.,.],[.,.]]],[[.,.],.]]
=> ? = 3
[8,7,4,5,6,3,2,1] => [1,8,2,7,3,4,5,6] => [8,1,7,2,6,5,4,3] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ? = 2
[8,6,4,5,7,3,2,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2
[7,6,4,5,8,3,2,1] => [1,7,2,6,3,4,5,8] => [8,2,7,3,6,5,4,1] => [[[.,.],[[.,.],[[[.,.],.],.]]],.]
=> ? = 2
[8,7,6,5,3,4,2,1] => [1,8,2,7,3,6,4,5] => [8,1,7,2,6,3,5,4] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[7,8,6,5,3,4,2,1] => [1,7,2,8,3,6,4,5] => [8,2,7,1,6,3,5,4] => [[[.,.],[.,.]],[[.,.],[[.,.],.]]]
=> ? = 3
[8,7,5,6,3,4,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3
[7,8,5,6,3,4,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3
[8,5,6,7,3,4,2,1] => [1,8,2,5,3,6,4,7] => [8,1,7,4,6,3,5,2] => [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> ? = 3
[5,6,7,8,3,4,2,1] => [1,5,3,7,2,6,4,8] => [8,4,6,2,7,3,5,1] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[8,7,5,4,3,6,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3
[7,8,5,4,3,6,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3
[8,7,4,5,3,6,2,1] => [1,8,2,7,3,4,5,6] => [8,1,7,2,6,5,4,3] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ? = 2
[8,7,5,3,4,6,2,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3
[7,8,5,3,4,6,2,1] => [1,7,2,8,3,5,4,6] => [8,2,7,1,6,4,5,3] => [[[.,.],[.,.]],[[[.,.],[.,.]],.]]
=> ? = 3
[8,7,4,3,5,6,2,1] => [1,8,2,7,3,4,5,6] => [8,1,7,2,6,5,4,3] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ? = 2
[8,7,3,4,5,6,2,1] => [1,8,2,7,3,4,5,6] => [8,1,7,2,6,5,4,3] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ? = 2
[7,8,3,4,5,6,2,1] => [1,7,2,8,3,4,5,6] => [8,2,7,1,6,5,4,3] => [[[.,.],[.,.]],[[[[.,.],.],.],.]]
=> ? = 2
[8,5,4,6,3,7,2,1] => [1,8,2,5,3,4,6,7] => [8,1,7,4,6,5,3,2] => [[.,.],[[[[.,.],[[.,.],.]],.],.]]
=> ? = 2
[8,3,4,5,6,7,2,1] => [1,8,2,3,4,5,6,7] => [8,1,7,6,5,4,3,2] => [[.,.],[[[[[[.,.],.],.],.],.],.]]
=> ? = 1
[7,5,4,6,3,8,2,1] => [1,7,2,5,3,4,6,8] => [8,2,7,4,6,5,3,1] => [[[.,.],[[[.,.],[[.,.],.]],.]],.]
=> ? = 2
[7,3,4,5,6,8,2,1] => [1,7,2,3,4,5,6,8] => [8,2,7,6,5,4,3,1] => [[[.,.],[[[[[.,.],.],.],.],.]],.]
=> ? = 1
[6,3,4,5,7,8,2,1] => [1,6,8,2,3,4,5,7] => [8,3,1,7,6,5,4,2] => [[[.,.],.],[[[[[.,.],.],.],.],.]]
=> ? = 1
[8,7,6,5,4,2,3,1] => [1,8,2,7,3,6,4,5] => [8,1,7,2,6,3,5,4] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[8,6,7,5,4,2,3,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3
[8,7,5,6,4,2,3,1] => [1,8,2,7,3,5,4,6] => [8,1,7,2,6,4,5,3] => [[.,.],[[.,.],[[[.,.],[.,.]],.]]]
=> ? = 3
[8,6,5,7,4,2,3,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3
[6,7,5,8,4,2,3,1] => [1,6,2,7,3,5,4,8] => [8,3,7,2,6,4,5,1] => [[[[.,.],[.,.]],[[.,.],[.,.]]],.]
=> ? = 3
[8,7,6,4,5,2,3,1] => [1,8,2,7,3,6,4,5] => [8,1,7,2,6,3,5,4] => [[.,.],[[.,.],[[.,.],[[.,.],.]]]]
=> ? = 3
[8,6,7,4,5,2,3,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3
[6,7,8,4,5,2,3,1] => [1,6,2,7,3,8,4,5] => [8,3,7,2,6,1,5,4] => [[[[.,.],[.,.]],[.,.]],[[.,.],.]]
=> ? = 3
[8,7,4,5,6,2,3,1] => [1,8,2,7,3,4,5,6] => [8,1,7,2,6,5,4,3] => [[.,.],[[.,.],[[[[.,.],.],.],.]]]
=> ? = 2
[8,6,4,5,7,2,3,1] => [1,8,2,6,3,4,5,7] => [8,1,7,3,6,5,4,2] => [[.,.],[[[.,.],[[[.,.],.],.]],.]]
=> ? = 2
[6,7,4,5,8,2,3,1] => [1,6,2,7,3,4,5,8] => [8,3,7,2,6,5,4,1] => [[[[.,.],[.,.]],[[[.,.],.],.]],.]
=> ? = 2
[8,6,7,5,3,2,4,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3
[8,6,5,7,3,2,4,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3
[8,5,6,7,3,2,4,1] => [1,8,2,5,3,6,4,7] => [8,1,7,4,6,3,5,2] => [[.,.],[[[[.,.],[.,.]],[.,.]],.]]
=> ? = 3
[8,6,7,5,2,3,4,1] => [1,8,2,6,3,7,4,5] => [8,1,7,3,6,2,5,4] => [[.,.],[[[.,.],[.,.]],[[.,.],.]]]
=> ? = 3
[8,6,5,7,2,3,4,1] => [1,8,2,6,3,5,4,7] => [8,1,7,3,6,4,5,2] => [[.,.],[[[.,.],[[.,.],[.,.]]],.]]
=> ? = 3
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 $2^{n-2}$.
The following 36 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001354The number of series nodes in the modular decomposition of a graph. St000353The number of inner valleys of a permutation. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000092The number of outer peaks of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001487The number of inner corners of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000632The jump number of the poset. St000640The rank of the largest boolean interval in a poset. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001624The breadth of a lattice. St001823The Stasinski-Voll length of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.
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!