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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000054
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => 1
[1,2] => [1,0,1,0]
=> [1,2] => [1,2] => 1
[2,1] => [1,1,0,0]
=> [2,1] => [2,1] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 3
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,2,1] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,2,1] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,3,1] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,3,2,1] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,3,2,1] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,4,2] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,3,4,2] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,4,3,2] => 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000382
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(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,2] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [3] => 3
[3,1,2] => [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,2] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,3] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,1] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [4] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,1] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => 1
[2,1,8,5,4,7,6,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[2,1,8,7,6,5,4,3,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[3,4,1,2,6,5,9,10,7,8] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? => ? = 4
[3,4,2,1,6,5,9,10,8,7] => [1,1,1,0,1,0,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> ? => ? = 4
[10,3,2,5,4,7,6,9,8,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,3,2,5,4,9,8,7,6,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,3,2,7,6,5,4,9,8,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,3,2,9,6,5,8,7,4,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,3,2,9,8,7,6,5,4,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,5,4,3,2,7,6,9,8,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,5,4,3,2,9,8,7,6,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,7,4,3,6,5,2,9,8,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,4,3,6,5,8,7,2,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,4,3,8,7,6,5,2,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,7,6,5,4,3,2,9,8,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,6,5,4,3,8,7,2,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,8,5,4,7,6,3,2,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,8,7,6,5,4,3,2,1,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[2,1,3,4,5,6,7,8,9,10,11] => [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] => ? = 2
[2,1,8,7,4,3,6,5,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[2,1,8,7,5,3,4,6,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[2,1,8,3,4,6,7,5,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[10,5,3,7,2,9,4,1,6,8,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,5,3,8,2,1,6,4,9,7,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,6,3,9,4,2,7,1,5,8,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,6,3,1,9,2,4,8,5,7,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,7,3,1,4,5,2,8,9,6,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,7,2,4,5,9,3,1,6,8,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,8,2,4,5,1,6,3,9,7,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,7,4,2,6,3,1,5,8,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,8,4,9,6,2,3,5,7,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,8,4,2,5,7,3,9,6,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,2,3,5,6,7,1,4,8,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,9,3,5,6,2,8,4,7,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,2,9,5,3,7,8,4,6,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,2,3,4,6,7,8,9,5,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[2,1,8,3,4,5,6,7,10,9] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,6,2] => ? = 2
[10,5,2,7,4,9,6,1,8,3,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,5,2,9,4,1,6,7,8,3,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,7,2,9,4,5,6,1,8,3,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,2,1,8,5,4,7,6,3,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,2,1,4,5,6,7,8,3,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,7,2,3,4,9,6,1,8,5,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,2,3,4,1,6,7,8,5,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,6,3,2,5,4,1,8,7,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,6,3,8,5,2,7,4,9,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,8,3,2,5,6,7,4,9,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,9,2,3,4,5,6,1,8,7,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,8,3,4,5,2,7,6,9,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
[10,1,2,7,4,3,6,5,8,9,12,11] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,2] => ? = 10
Description
The first part of an integer composition.
Matching statistic: St000505
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> {{1}}
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 4
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[7,3,2,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7
[7,2,3,1,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7
[7,3,1,2,4,5,6,8] => [7,4,3,2,5,6,1,8] => ?
=> ?
=> ? = 7
[7,2,1,3,4,5,6,8] => [7,3,2,4,5,6,1,8] => ?
=> ?
=> ? = 7
[6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,4,6,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[4,5,6,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,5,3,4,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,3,4,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,3,4,5,2,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,4,6,2,3,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,3,2,4,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,5,2,3,4,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,2,3,4,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,3,2,5,1,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,2,3,4,5,1,7,8] => [6,5,3,4,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,5,4,3,1,2,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,4,3,1,2,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,5,3,1,2,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,4,6,3,1,2,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,3,4,1,2,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,3,5,1,2,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,5,4,2,1,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,4,2,1,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,5,2,1,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,4,6,2,1,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[4,5,6,2,1,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,5,4,1,2,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[4,5,6,1,2,3,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,5,3,2,1,4,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,3,2,1,4,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,6,2,1,3,4,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,5,1,2,3,4,7,8] => [6,5,3,4,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,3,2,1,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,3,4,2,1,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,3,1,2,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,2,1,3,5,7,8] => [6,5,4,3,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,4,1,2,3,5,7,8] => [6,5,3,4,2,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,3,2,1,4,5,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,2,3,1,4,5,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,3,1,2,4,5,7,8] => [6,4,3,2,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[6,2,1,3,4,5,7,8] => [6,3,2,4,5,1,7,8] => ?
=> ?
=> ? = 6
[6,1,2,3,4,5,7,8] => [6,2,3,4,5,1,7,8] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 6
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[4,5,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[5,3,4,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[4,3,5,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[3,4,5,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
[5,4,2,3,1,6,7,8] => [5,4,3,2,1,6,7,8] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 5
Description
The biggest entry in the block containing the 1.
Matching statistic: St000326
Mp00114: Permutations —connectivity set⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
St000326: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1] => => ? = 1
[1,2] => 1 => 1
[2,1] => 0 => 2
[1,2,3] => 11 => 1
[1,3,2] => 10 => 1
[2,1,3] => 01 => 2
[2,3,1] => 00 => 3
[3,1,2] => 00 => 3
[3,2,1] => 00 => 3
[1,2,3,4] => 111 => 1
[1,2,4,3] => 110 => 1
[1,3,2,4] => 101 => 1
[1,3,4,2] => 100 => 1
[1,4,2,3] => 100 => 1
[1,4,3,2] => 100 => 1
[2,1,3,4] => 011 => 2
[2,1,4,3] => 010 => 2
[2,3,1,4] => 001 => 3
[2,3,4,1] => 000 => 4
[2,4,1,3] => 000 => 4
[2,4,3,1] => 000 => 4
[3,1,2,4] => 001 => 3
[3,1,4,2] => 000 => 4
[3,2,1,4] => 001 => 3
[3,2,4,1] => 000 => 4
[3,4,1,2] => 000 => 4
[3,4,2,1] => 000 => 4
[4,1,2,3] => 000 => 4
[4,1,3,2] => 000 => 4
[4,2,1,3] => 000 => 4
[4,2,3,1] => 000 => 4
[4,3,1,2] => 000 => 4
[4,3,2,1] => 000 => 4
[1,2,3,4,5] => 1111 => 1
[1,2,3,5,4] => 1110 => 1
[1,2,4,3,5] => 1101 => 1
[1,2,4,5,3] => 1100 => 1
[1,2,5,3,4] => 1100 => 1
[1,2,5,4,3] => 1100 => 1
[1,3,2,4,5] => 1011 => 1
[1,3,2,5,4] => 1010 => 1
[1,3,4,2,5] => 1001 => 1
[1,3,4,5,2] => 1000 => 1
[1,3,5,2,4] => 1000 => 1
[1,3,5,4,2] => 1000 => 1
[1,4,2,3,5] => 1001 => 1
[1,4,2,5,3] => 1000 => 1
[1,4,3,2,5] => 1001 => 1
[1,4,3,5,2] => 1000 => 1
[1,4,5,2,3] => 1000 => 1
[1,4,5,3,2] => 1000 => 1
[8,6,7,5,4,3,2,1] => ? => ? = 8
[7,6,8,4,5,3,2,1] => ? => ? = 8
[6,7,8,4,5,3,2,1] => ? => ? = 8
[8,6,5,4,7,3,2,1] => ? => ? = 8
[8,5,6,4,7,3,2,1] => ? => ? = 8
[8,6,4,5,7,3,2,1] => ? => ? = 8
[8,5,4,6,7,3,2,1] => ? => ? = 8
[8,4,5,6,7,3,2,1] => ? => ? = 8
[8,7,6,4,3,5,2,1] => ? => ? = 8
[7,8,6,4,3,5,2,1] => ? => ? = 8
[8,6,7,4,3,5,2,1] => ? => ? = 8
[6,7,8,4,3,5,2,1] => ? => ? = 8
[7,8,3,4,5,6,2,1] => ? => ? = 8
[8,5,4,6,3,7,2,1] => ? => ? = 8
[8,6,5,7,4,2,3,1] => ? => ? = 8
[7,5,6,8,4,2,3,1] => ? => ? = 8
[8,7,5,4,6,2,3,1] => ? => ? = 8
[8,7,4,5,6,2,3,1] => ? => ? = 8
[8,6,5,4,7,2,3,1] => ? => ? = 8
[8,5,6,4,7,2,3,1] => ? => ? = 8
[8,6,4,5,7,2,3,1] => ? => ? = 8
[8,4,5,6,7,2,3,1] => ? => ? = 8
[8,7,6,5,3,2,4,1] => ? => ? = 8
[8,6,7,5,3,2,4,1] => ? => ? = 8
[8,7,5,6,3,2,4,1] => ? => ? = 8
[7,8,5,6,2,3,4,1] => ? => ? = 8
[8,7,6,4,3,2,5,1] => ? => ? = 8
[7,8,6,4,3,2,5,1] => ? => ? = 8
[8,6,7,4,3,2,5,1] => ? => ? = 8
[6,7,8,4,3,2,5,1] => ? => ? = 8
[8,7,6,3,4,2,5,1] => ? => ? = 8
[8,6,7,3,4,2,5,1] => ? => ? = 8
[7,6,8,3,4,2,5,1] => ? => ? = 8
[7,6,8,4,2,3,5,1] => ? => ? = 8
[7,8,6,3,2,4,5,1] => ? => ? = 8
[8,7,5,4,3,2,6,1] => ? => ? = 8
[8,7,5,3,4,2,6,1] => ? => ? = 8
[7,8,5,3,4,2,6,1] => ? => ? = 8
[8,7,5,3,2,4,6,1] => ? => ? = 8
[7,8,3,4,2,5,6,1] => ? => ? = 8
[7,8,4,2,3,5,6,1] => ? => ? = 8
[8,6,5,3,4,2,7,1] => ? => ? = 8
[8,5,6,3,4,2,7,1] => ? => ? = 8
[8,4,3,5,6,2,7,1] => ? => ? = 8
[8,5,6,4,2,3,7,1] => ? => ? = 8
[8,6,2,3,4,5,7,1] => ? => ? = 8
[8,3,4,5,2,6,7,1] => ? => ? = 8
[8,5,4,2,3,6,7,1] => ? => ? = 8
[8,5,2,3,4,6,7,1] => ? => ? = 8
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 100%
Values
[1] => => => ? = 1 - 1
[1,2] => 1 => 0 => 0 = 1 - 1
[2,1] => 0 => 1 => 1 = 2 - 1
[1,2,3] => 11 => 00 => 0 = 1 - 1
[1,3,2] => 10 => 01 => 0 = 1 - 1
[2,1,3] => 01 => 10 => 1 = 2 - 1
[2,3,1] => 00 => 11 => 2 = 3 - 1
[3,1,2] => 00 => 11 => 2 = 3 - 1
[3,2,1] => 00 => 11 => 2 = 3 - 1
[1,2,3,4] => 111 => 000 => 0 = 1 - 1
[1,2,4,3] => 110 => 001 => 0 = 1 - 1
[1,3,2,4] => 101 => 010 => 0 = 1 - 1
[1,3,4,2] => 100 => 011 => 0 = 1 - 1
[1,4,2,3] => 100 => 011 => 0 = 1 - 1
[1,4,3,2] => 100 => 011 => 0 = 1 - 1
[2,1,3,4] => 011 => 100 => 1 = 2 - 1
[2,1,4,3] => 010 => 101 => 1 = 2 - 1
[2,3,1,4] => 001 => 110 => 2 = 3 - 1
[2,3,4,1] => 000 => 111 => 3 = 4 - 1
[2,4,1,3] => 000 => 111 => 3 = 4 - 1
[2,4,3,1] => 000 => 111 => 3 = 4 - 1
[3,1,2,4] => 001 => 110 => 2 = 3 - 1
[3,1,4,2] => 000 => 111 => 3 = 4 - 1
[3,2,1,4] => 001 => 110 => 2 = 3 - 1
[3,2,4,1] => 000 => 111 => 3 = 4 - 1
[3,4,1,2] => 000 => 111 => 3 = 4 - 1
[3,4,2,1] => 000 => 111 => 3 = 4 - 1
[4,1,2,3] => 000 => 111 => 3 = 4 - 1
[4,1,3,2] => 000 => 111 => 3 = 4 - 1
[4,2,1,3] => 000 => 111 => 3 = 4 - 1
[4,2,3,1] => 000 => 111 => 3 = 4 - 1
[4,3,1,2] => 000 => 111 => 3 = 4 - 1
[4,3,2,1] => 000 => 111 => 3 = 4 - 1
[1,2,3,4,5] => 1111 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 1110 => 0001 => 0 = 1 - 1
[1,2,4,3,5] => 1101 => 0010 => 0 = 1 - 1
[1,2,4,5,3] => 1100 => 0011 => 0 = 1 - 1
[1,2,5,3,4] => 1100 => 0011 => 0 = 1 - 1
[1,2,5,4,3] => 1100 => 0011 => 0 = 1 - 1
[1,3,2,4,5] => 1011 => 0100 => 0 = 1 - 1
[1,3,2,5,4] => 1010 => 0101 => 0 = 1 - 1
[1,3,4,2,5] => 1001 => 0110 => 0 = 1 - 1
[1,3,4,5,2] => 1000 => 0111 => 0 = 1 - 1
[1,3,5,2,4] => 1000 => 0111 => 0 = 1 - 1
[1,3,5,4,2] => 1000 => 0111 => 0 = 1 - 1
[1,4,2,3,5] => 1001 => 0110 => 0 = 1 - 1
[1,4,2,5,3] => 1000 => 0111 => 0 = 1 - 1
[1,4,3,2,5] => 1001 => 0110 => 0 = 1 - 1
[1,4,3,5,2] => 1000 => 0111 => 0 = 1 - 1
[1,4,5,2,3] => 1000 => 0111 => 0 = 1 - 1
[1,4,5,3,2] => 1000 => 0111 => 0 = 1 - 1
[8,6,7,5,4,3,2,1] => ? => ? => ? = 8 - 1
[7,6,8,4,5,3,2,1] => ? => ? => ? = 8 - 1
[6,7,8,4,5,3,2,1] => ? => ? => ? = 8 - 1
[8,6,5,4,7,3,2,1] => ? => ? => ? = 8 - 1
[8,5,6,4,7,3,2,1] => ? => ? => ? = 8 - 1
[8,6,4,5,7,3,2,1] => ? => ? => ? = 8 - 1
[8,5,4,6,7,3,2,1] => ? => ? => ? = 8 - 1
[8,4,5,6,7,3,2,1] => ? => ? => ? = 8 - 1
[8,7,6,4,3,5,2,1] => ? => ? => ? = 8 - 1
[7,8,6,4,3,5,2,1] => ? => ? => ? = 8 - 1
[8,6,7,4,3,5,2,1] => ? => ? => ? = 8 - 1
[6,7,8,4,3,5,2,1] => ? => ? => ? = 8 - 1
[7,8,3,4,5,6,2,1] => ? => ? => ? = 8 - 1
[8,5,4,6,3,7,2,1] => ? => ? => ? = 8 - 1
[8,6,5,7,4,2,3,1] => ? => ? => ? = 8 - 1
[7,5,6,8,4,2,3,1] => ? => ? => ? = 8 - 1
[8,7,5,4,6,2,3,1] => ? => ? => ? = 8 - 1
[8,7,4,5,6,2,3,1] => ? => ? => ? = 8 - 1
[8,6,5,4,7,2,3,1] => ? => ? => ? = 8 - 1
[8,5,6,4,7,2,3,1] => ? => ? => ? = 8 - 1
[8,6,4,5,7,2,3,1] => ? => ? => ? = 8 - 1
[8,4,5,6,7,2,3,1] => ? => ? => ? = 8 - 1
[8,7,6,5,3,2,4,1] => ? => ? => ? = 8 - 1
[8,6,7,5,3,2,4,1] => ? => ? => ? = 8 - 1
[8,7,5,6,3,2,4,1] => ? => ? => ? = 8 - 1
[7,8,5,6,2,3,4,1] => ? => ? => ? = 8 - 1
[8,7,6,4,3,2,5,1] => ? => ? => ? = 8 - 1
[7,8,6,4,3,2,5,1] => ? => ? => ? = 8 - 1
[8,6,7,4,3,2,5,1] => ? => ? => ? = 8 - 1
[6,7,8,4,3,2,5,1] => ? => ? => ? = 8 - 1
[8,7,6,3,4,2,5,1] => ? => ? => ? = 8 - 1
[8,6,7,3,4,2,5,1] => ? => ? => ? = 8 - 1
[7,6,8,3,4,2,5,1] => ? => ? => ? = 8 - 1
[7,6,8,4,2,3,5,1] => ? => ? => ? = 8 - 1
[7,8,6,3,2,4,5,1] => ? => ? => ? = 8 - 1
[8,7,5,4,3,2,6,1] => ? => ? => ? = 8 - 1
[8,7,5,3,4,2,6,1] => ? => ? => ? = 8 - 1
[7,8,5,3,4,2,6,1] => ? => ? => ? = 8 - 1
[8,7,5,3,2,4,6,1] => ? => ? => ? = 8 - 1
[7,8,3,4,2,5,6,1] => ? => ? => ? = 8 - 1
[7,8,4,2,3,5,6,1] => ? => ? => ? = 8 - 1
[8,6,5,3,4,2,7,1] => ? => ? => ? = 8 - 1
[8,5,6,3,4,2,7,1] => ? => ? => ? = 8 - 1
[8,4,3,5,6,2,7,1] => ? => ? => ? = 8 - 1
[8,5,6,4,2,3,7,1] => ? => ? => ? = 8 - 1
[8,6,2,3,4,5,7,1] => ? => ? => ? = 8 - 1
[8,3,4,5,2,6,7,1] => ? => ? => ? = 8 - 1
[8,5,4,2,3,6,7,1] => ? => ? => ? = 8 - 1
[8,5,2,3,4,6,7,1] => ? => ? => ? = 8 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000439
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => => [1] => [1,0]
=> 2 = 1 + 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[8,6,7,5,4,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[7,6,8,4,5,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[6,7,8,4,5,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,5,4,7,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,6,4,7,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,4,5,7,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,4,6,7,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,4,5,6,7,3,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,6,4,3,5,2,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,6,4,3,5,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,7,4,3,5,2,1] => ? => ? => ?
=> ? = 8 + 1
[6,7,8,4,3,5,2,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,3,4,5,6,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,4,6,3,7,2,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,5,7,4,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[7,5,6,8,4,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,5,4,6,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,4,5,6,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,5,4,7,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,6,4,7,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,4,5,7,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[8,4,5,6,7,2,3,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,6,5,3,2,4,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,7,5,3,2,4,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,5,6,3,2,4,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,5,6,2,3,4,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,6,4,3,2,5,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,6,4,3,2,5,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,7,4,3,2,5,1] => ? => ? => ?
=> ? = 8 + 1
[6,7,8,4,3,2,5,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,6,3,4,2,5,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,7,3,4,2,5,1] => ? => ? => ?
=> ? = 8 + 1
[7,6,8,3,4,2,5,1] => ? => ? => ?
=> ? = 8 + 1
[7,6,8,4,2,3,5,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,6,3,2,4,5,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,5,4,3,2,6,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,5,3,4,2,6,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,5,3,4,2,6,1] => ? => ? => ?
=> ? = 8 + 1
[8,7,5,3,2,4,6,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,3,4,2,5,6,1] => ? => ? => ?
=> ? = 8 + 1
[7,8,4,2,3,5,6,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,5,3,4,2,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,6,3,4,2,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,4,3,5,6,2,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,6,4,2,3,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,6,2,3,4,5,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,3,4,5,2,6,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,4,2,3,6,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,5,2,3,4,6,7,1] => ? => ? => ?
=> ? = 8 + 1
[8,4,2,3,5,6,7,1] => ? => ? => ?
=> ? = 8 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => 1
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1,1] => 1
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> [2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 3
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [3] => 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 4
[1,2,3,4,5] => [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
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [8] => ? = 8
[7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[6,5,7,8,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [8] => ? = 8
[5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [8] => ? = 8
[8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,6,8,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,5,6,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,4,5,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,5,4,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,6,8,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,6,5,8,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [8] => ? = 8
[6,5,7,8,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [8] => ? = 8
[5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [8] => ? = 8
[8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,7,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,6,8,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[6,7,8,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,6,7,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,6,8,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
[7,8,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ? = 8
[8,7,4,5,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ? = 8
Description
The last part of an integer composition.
Matching statistic: St000066
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 1
[1,2] => [1,0,1,0]
=> [1,2] => [[1,0],[0,1]]
=> 1
[2,1] => [1,1,0,0]
=> [2,1] => [[0,1],[1,0]]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> 1
[1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,2,3,4,5,7,6] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> ? = 1
[1,2,3,4,6,5,7] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? = 1
[1,3,4,6,7,2,5] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,4,6,7,5,2] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,4,7,2,5,6] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,4,7,2,6,5] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,4,7,5,2,6] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,4,7,5,6,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,4,7,6,2,5] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,4,7,6,5,2] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0]]
=> ? = 1
[1,3,5,2,7,4,6] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[1,3,5,2,7,6,4] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[1,3,5,4,7,2,6] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[1,3,5,4,7,6,2] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0]]
=> ? = 1
[1,3,5,6,2,7,4] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,6,4,5,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,5,6,4,7,2] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,6,4,5,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,5,6,7,2,4] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,5,6,7,4,2] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,7,4,5,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,5,7,2,4,6] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,5,7,2,6,4] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,5,7,4,2,6] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,5,7,4,6,2] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,5,7,6,2,4] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,5,7,6,4,2] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,3,7,4,6,5,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,2,4,7,5] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,6,2,5,7,4] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,6,2,7,4,5] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,2,7,5,4] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,4,2,7,5] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,6,4,5,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,6,4,7,2,5] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,4,7,5,2] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,5,2,7,4] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,6,5,4,7,2] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,0,1,0]]
=> ? = 1
[1,3,6,5,7,2,4] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,5,7,4,2] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,7,5,4,6,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,7,2,4,5] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,7,2,5,4] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,7,4,2,5] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,7,4,5,2] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,7,5,2,4] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,6,7,5,4,2] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,3,7,5,6,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,7,2,4,5,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,7,2,4,6,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,7,2,5,4,6] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,7,2,5,6,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,7,2,6,4,5] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
[1,3,7,2,6,5,4] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,3,7,6,5,4,2] => [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,1,0,0],[0,0,0,1,0,0,0],[0,0,1,0,0,0,0]]
=> ? = 1
Description
The column of the unique '1' in the first row of the alternating sign matrix.
The generating function of this statistic is given by
$$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$$
see [2].
Matching statistic: St000026
(load all 109 compositions to match this statistic)
(load all 109 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 70%
St000026: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 70%
Values
[1] => [1,0]
=> 1
[1,2] => [1,0,1,0]
=> 1
[2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [1,1,0,1,0,0]
=> 3
[3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 4
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 4
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 4
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 4
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,5,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,5,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,5,4,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8
[6,7,8,5,4,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 8
[8,7,5,6,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,5,6,4,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,6,5,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,5,6,7,4,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,5,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 8
[6,7,5,8,4,3,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 8
[7,5,6,8,4,3,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 8
[6,5,7,8,4,3,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 8
[5,6,7,8,4,3,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 8
[8,7,6,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,4,5,3,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,4,5,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,4,5,3,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8
[6,7,8,4,5,3,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 8
[8,7,5,4,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,4,5,6,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,5,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,5,6,4,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,4,5,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,5,4,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,4,5,6,7,3,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,6,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,5,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,5,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,5,3,4,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8
[8,7,5,6,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,5,6,3,4,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,5,6,7,3,4,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,5,8,3,4,2,1] => [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 8
[6,7,5,8,3,4,2,1] => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 8
[6,5,7,8,3,4,2,1] => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> ? = 8
[5,6,7,8,3,4,2,1] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 8
[8,7,6,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,4,3,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,4,3,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,4,3,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8
[6,7,8,4,3,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 8
[8,7,6,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,3,4,5,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,3,4,5,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,3,4,5,2,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8
[6,7,8,3,4,5,2,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 8
[8,7,5,4,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,5,4,3,6,2,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8
[8,7,4,5,3,6,2,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The position of the first return of a Dyck path.
Matching statistic: St000025
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 70%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 70%
Values
[1] => => [1] => [1,0]
=> 1
[1,2] => 1 => [1,1] => [1,0,1,0]
=> 1
[2,1] => 0 => [2] => [1,1,0,0]
=> 2
[1,2,3] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => 01 => [2,1] => [1,1,0,0,1,0]
=> 2
[2,3,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[3,1,2] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[3,2,1] => 00 => [3] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => 011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => 010 => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[2,3,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,2,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,1,4,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,1,4] => 001 => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,3,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,1,3] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,2,3,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,1,2] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[4,3,2,1] => 000 => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,3,4,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,2,3,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,2,5,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[8,7,6,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,5,4,3,2,1] => ? => ? => ?
=> ? = 8
[7,6,8,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,7,8,5,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,5,6,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,5,6,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,5,7,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,5,6,7,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,5,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,7,5,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,5,6,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,5,7,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[5,6,7,8,4,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,6,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,4,5,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,4,5,3,2,1] => ? => ? => ?
=> ? = 8
[6,7,8,4,5,3,2,1] => ? => ? => ?
=> ? = 8
[8,7,5,4,6,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,4,5,6,3,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,5,4,7,3,2,1] => ? => ? => ?
=> ? = 8
[8,5,6,4,7,3,2,1] => ? => ? => ?
=> ? = 8
[8,6,4,5,7,3,2,1] => ? => ? => ?
=> ? = 8
[8,5,4,6,7,3,2,1] => ? => ? => ?
=> ? = 8
[8,4,5,6,7,3,2,1] => ? => ? => ?
=> ? = 8
[8,7,6,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,5,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,5,6,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,5,6,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,5,6,7,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,5,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,7,5,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,5,7,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[5,6,7,8,3,4,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,6,4,3,5,2,1] => ? => ? => ?
=> ? = 8
[7,8,6,4,3,5,2,1] => ? => ? => ?
=> ? = 8
[8,6,7,4,3,5,2,1] => ? => ? => ?
=> ? = 8
[7,6,8,4,3,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,7,8,4,3,5,2,1] => ? => ? => ?
=> ? = 8
[8,7,6,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,6,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,6,7,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,6,8,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[6,7,8,3,4,5,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,5,4,3,6,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[7,8,5,4,3,6,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
[8,7,4,5,3,6,2,1] => 0000000 => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000740The last entry of a permutation. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000051The size of the left subtree of a binary tree. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000060The greater neighbor of the maximum. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix.
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