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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St000054
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00325: Permutations —ones to leading⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,1,3] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,1,4,3] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,4,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,4,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,2,4] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,3,2,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,3,1,4] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,2,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [2,3,4,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,4,1,2] => 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,3,2] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,1,5,4,3] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,5,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,5,3,4,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [3,1,2,4,5] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,4,2,3,5] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,4,3,1,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [2,5,3,1,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,2,4] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,4,2,5,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,4,3,5,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [3,5,4,1,2] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,1,3,5,2] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,3,2,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,3,1,5,4] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,2,5,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [2,3,5,4,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [3,4,1,2,5] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,2,4,3,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,4,1,5,3] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,5,2,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,1,5,4,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,4,5,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,4,5,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,1,3,2,5] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,4,2,5] => 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: St000011
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [5,1,6,2,3,8,4,7] => [1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [5,6,1,2,3,8,4,7] => [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> ? = 3
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000439
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2 = 1 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 3 + 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 3 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000678
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? = 3
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001050
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001050: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> {{1}}
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5},{6,7,8}}
=> ? = 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0,1,0]
=> {{1,5,7},{2,3},{4},{6},{8}}
=> ? = 3
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> {{1},{2,5,7},{3},{4},{6},{8}}
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> {{1,5,7},{2},{3},{4},{6},{8}}
=> ? = 3
Description
The number of terminal closers of a set partition.
A closer of a set partition is a number that is maximal in its block. In particular, a singleton is a closer. This statistic counts the number of terminal closers. In other words, this is the number of closers such that all larger elements are also closers.
Matching statistic: St000971
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => {{1}}
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => {{1,3},{2}}
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => {{1,2,3}}
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 3
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => {{1,4},{2},{3},{5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
=> [1,2,7,3,4,8,5,6] => ?
=> ? = 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3,7,8,6] => ?
=> ? = 3
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0,1,1,0,0]
=> [4,6,1,2,3,7,8,5] => ?
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,1,0,0]
=> [4,6,1,2,3,5,7,8] => {{1,4},{2,6},{3},{5},{7},{8}}
=> ? = 3
[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]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St000025
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 3
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St001784
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001784: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001784: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => {{1}}
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => {{1,2}}
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => {{1},{2}}
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => {{1,2,3}}
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => {{1,2},{3}}
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => {{1,3},{2}}
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => {{1},{2},{3}}
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => {{1,2,3,4}}
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => {{1,2,3,4,5,6,7,8}}
=> ? = 2
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 1
[1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [3,1,6,7,2,4,5,8] => {{1,3,6},{2},{4,7},{5},{8}}
=> ? = 3
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [3,1,6,8,2,4,5,7] => {{1,3,6},{2},{4,8},{5},{7}}
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [3,1,7,8,2,4,5,6] => {{1,3,7},{2},{4,8},{5},{6}}
=> ? = 3
[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]
=> [1,2,3,4,5,6,7,8] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
Description
The minimum of the smallest closer and the second element of the block containing 1 in a set partition.
A closer of a set partition is the maximal element of a non-singleton block. This statistic is defined as $1$ if $\{1\}$ is a singleton block, and otherwise the minimum of the smallest closer and the second element of the block containing $1$.
Matching statistic: St000476
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000476: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,6,5,7,8] => [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
[1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,5,6,2,3,8,4,7] => [7,4,8,3,2,6,5,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [5,1,6,2,3,8,4,7] => [7,4,8,3,2,6,1,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [5,6,1,2,3,8,4,7] => [7,4,8,3,2,1,6,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => [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
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path.
For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which
is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is
$$
\sum_v (j_v-i_v)/2.
$$
Matching statistic: St000947
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1 - 1
[1,0,1,0]
=> [2,1] => [1,2] => [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,6,5,7,8] => [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
[1,1,1,1,0,0,1,0,1,1,0,0,0,1,0,0]
=> [1,5,6,2,3,8,4,7] => [7,4,8,3,2,6,5,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,0,0,1,1,0,0,0,1,0,0]
=> [5,1,6,2,3,8,4,7] => [7,4,8,3,2,6,1,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,1,0,0]
=> [5,6,1,2,3,8,4,7] => [7,4,8,3,2,1,6,5] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 3 - 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => [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
Description
The major index east count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000068The number of minimal elements in a poset. St000007The number of saliances of the permutation. St000989The number of final rises 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. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000738The first entry in the last row of a standard tableau. St000654The first descent of a permutation. St000740The last entry of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000133The "bounce" of a permutation. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000005The bounce statistic of a Dyck path. St000051The size of the left subtree of a binary tree. St000120The number of left tunnels of a Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000260The radius of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St000383The last part of an integer composition. St001621The number of atoms of a lattice.
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