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Your data matches 67 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)
Mp00277: Permutations —catalanization⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 1
[2,1] => [2,1] => 2
[1,2,3] => [1,2,3] => 1
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 2
[2,3,1] => [2,3,1] => 2
[3,1,2] => [2,3,1] => 2
[3,2,1] => [3,2,1] => 3
[1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => 1
[1,4,2,3] => [1,3,4,2] => 1
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 2
[2,3,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [2,3,4,1] => 2
[2,4,1,3] => [4,3,1,2] => 4
[2,4,3,1] => [2,4,3,1] => 2
[3,1,2,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,3,4,1] => 2
[3,2,1,4] => [3,2,1,4] => 3
[3,2,4,1] => [3,2,4,1] => 3
[3,4,1,2] => [4,3,2,1] => 4
[3,4,2,1] => [3,4,2,1] => 3
[4,1,2,3] => [2,3,4,1] => 2
[4,1,3,2] => [2,4,3,1] => 2
[4,2,1,3] => [3,2,4,1] => 3
[4,2,3,1] => [3,4,2,1] => 3
[4,3,1,2] => [3,4,2,1] => 3
[4,3,2,1] => [4,3,2,1] => 4
[1,2,3,4,5] => [1,2,3,4,5] => 1
[1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => 1
[1,2,5,3,4] => [1,2,4,5,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => 1
[1,3,4,2,5] => [1,3,4,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => 1
[1,3,5,2,4] => [1,5,4,2,3] => 1
[1,3,5,4,2] => [1,3,5,4,2] => 1
[1,4,2,3,5] => [1,3,4,2,5] => 1
[1,4,2,5,3] => [1,3,4,5,2] => 1
[1,4,3,2,5] => [1,4,3,2,5] => 1
[1,4,3,5,2] => [1,4,3,5,2] => 1
[1,4,5,2,3] => [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: St000025
Mp00277: Permutations —catalanization⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,5,4,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 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: St000740
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1] => [2,3,1] => [1,3,2] => 2
[3,1,2] => [2,3,1] => [1,3,2] => 2
[3,2,1] => [3,2,1] => [1,2,3] => 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => 1
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [2,3,1,4] => [4,1,3,2] => 2
[2,3,4,1] => [2,3,4,1] => [1,4,3,2] => 2
[2,4,1,3] => [4,3,1,2] => [2,1,3,4] => 4
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 2
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => 2
[3,1,4,2] => [2,3,4,1] => [1,4,3,2] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 3
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 4
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 3
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => 2
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => 2
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => 3
[4,2,3,1] => [3,4,2,1] => [1,2,4,3] => 3
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [3,5,4,2,1] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 1
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => 1
[1,3,5,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => 1
[1,3,5,4,2] => [1,3,5,4,2] => [2,4,5,3,1] => 1
[1,4,2,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => 1
[1,4,2,5,3] => [1,3,4,5,2] => [2,5,4,3,1] => 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 1
[1,4,3,5,2] => [1,4,3,5,2] => [2,5,3,4,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001184
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[3,1,4,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,5,3] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 4
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St000051
Mp00277: Permutations —catalanization⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> 0 = 1 - 1
[1,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[2,1] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,3,1] => [2,3,1] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,1,2] => [2,3,1] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,4,2,3] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[2,3,4,1] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[2,4,1,3] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> 3 = 4 - 1
[2,4,3,1] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[3,1,2,4] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[3,1,4,2] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[3,2,4,1] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[3,4,1,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 3 = 4 - 1
[3,4,2,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[4,1,2,3] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[4,1,3,2] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[4,2,1,3] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[4,2,3,1] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[4,3,1,2] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> 2 = 3 - 1
[4,3,2,1] => [4,3,2,1] => [[[[.,.],.],.],.]
=> 3 = 4 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> 0 = 1 - 1
[1,3,5,4,2] => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,4,2,5,3] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> 0 = 1 - 1
Description
The size of the left subtree of a binary tree.
Matching statistic: St000439
Mp00277: Permutations —catalanization⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2 = 1 + 1
[1,2] => [1,2] => [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,4,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
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: St000971
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> 1
[1,2] => [1,2] => [1,2] => {{1},{2}}
=> 1
[2,1] => [2,1] => [2,1] => {{1,2}}
=> 2
[1,2,3] => [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 1
[1,3,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,3,1] => [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,1,2] => [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,2,1] => [3,2,1] => [3,1,2] => {{1,2,3}}
=> 3
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[2,3,4,1] => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[2,4,1,3] => [4,3,1,2] => [2,4,1,3] => {{1,2,3,4}}
=> 4
[2,4,3,1] => [2,4,3,1] => [4,2,1,3] => {{1,3,4},{2}}
=> 2
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,1,4,2] => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 3
[3,2,4,1] => [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[3,4,1,2] => [4,3,2,1] => [4,1,2,3] => {{1,2,3,4}}
=> 4
[3,4,2,1] => [3,4,2,1] => [4,1,3,2] => {{1,2,4},{3}}
=> 3
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 2
[4,1,3,2] => [2,4,3,1] => [4,2,1,3] => {{1,3,4},{2}}
=> 2
[4,2,1,3] => [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[4,2,3,1] => [3,4,2,1] => [4,1,3,2] => {{1,2,4},{3}}
=> 3
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => {{1,2,4},{3}}
=> 3
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => {{1,2,3,4}}
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,3,5,2,4] => {{1},{2,3,4,5}}
=> 1
[1,3,5,4,2] => [1,3,5,4,2] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
[1,4,2,5,3] => [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 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: St001201
Mp00277: Permutations —catalanization⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001201: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 1
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[2,4,1,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,1,4,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[4,2,3,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [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
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,3,5,4,2] => [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path.
Matching statistic: St001291
Mp00277: Permutations —catalanization⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,2] => [1,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,1,2] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,3,1,4] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[3,1,2,4] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[4,2,1,3] => [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[4,2,3,1] => [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[4,3,1,2] => [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,3,4] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,3,5,2,4] => [1,5,4,2,3] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,3,5,4,2] => [1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,4,2,3,5] => [1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,5,3] => [1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
The following 57 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000133The "bounce" of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000678The number of up steps after the last double rise of a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St001060The distinguishing index of a graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001498The normalised height of a Nakayama algebra with magnitude 1. St000259The diameter of a connected graph. St001875The number of simple modules with projective dimension at most 1. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000264The girth of a graph, which is not a tree. St001330The hat guessing number of a graph. St001645The pebbling number of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000352The Elizalde-Pak rank of a permutation. St000366The number of double descents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001621The number of atoms of a lattice.
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