Your data matches 61 different statistics following compositions of up to 3 maps.
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Mp00090: Permutations cycle-as-one-line notationPermutations
St000740: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,2] => [1,2] => 2
[2,1] => [1,2] => 2
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,2,3] => 3
[2,1,3] => [1,2,3] => 3
[2,3,1] => [1,2,3] => 3
[3,1,2] => [1,3,2] => 2
[3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => 4
[1,2,4,3] => [1,2,3,4] => 4
[1,3,2,4] => [1,2,3,4] => 4
[1,3,4,2] => [1,2,3,4] => 4
[1,4,2,3] => [1,2,4,3] => 3
[1,4,3,2] => [1,2,4,3] => 3
[2,1,3,4] => [1,2,3,4] => 4
[2,1,4,3] => [1,2,3,4] => 4
[2,3,1,4] => [1,2,3,4] => 4
[2,3,4,1] => [1,2,3,4] => 4
[2,4,1,3] => [1,2,4,3] => 3
[2,4,3,1] => [1,2,4,3] => 3
[3,1,2,4] => [1,3,2,4] => 4
[3,1,4,2] => [1,3,4,2] => 2
[3,2,1,4] => [1,3,2,4] => 4
[3,2,4,1] => [1,3,4,2] => 2
[3,4,1,2] => [1,3,2,4] => 4
[3,4,2,1] => [1,3,2,4] => 4
[4,1,2,3] => [1,4,3,2] => 2
[4,1,3,2] => [1,4,2,3] => 3
[4,2,1,3] => [1,4,3,2] => 2
[4,2,3,1] => [1,4,2,3] => 3
[4,3,1,2] => [1,4,2,3] => 3
[4,3,2,1] => [1,4,2,3] => 3
[1,2,3,4,5] => [1,2,3,4,5] => 5
[1,2,3,5,4] => [1,2,3,4,5] => 5
[1,2,4,3,5] => [1,2,3,4,5] => 5
[1,2,4,5,3] => [1,2,3,4,5] => 5
[1,2,5,3,4] => [1,2,3,5,4] => 4
[1,2,5,4,3] => [1,2,3,5,4] => 4
[1,3,2,4,5] => [1,2,3,4,5] => 5
[1,3,2,5,4] => [1,2,3,4,5] => 5
[1,3,4,2,5] => [1,2,3,4,5] => 5
[1,3,4,5,2] => [1,2,3,4,5] => 5
[1,3,5,2,4] => [1,2,3,5,4] => 4
[1,3,5,4,2] => [1,2,3,5,4] => 4
[1,4,2,3,5] => [1,2,4,3,5] => 5
[1,4,2,5,3] => [1,2,4,5,3] => 3
[1,4,3,2,5] => [1,2,4,3,5] => 5
[1,4,3,5,2] => [1,2,4,5,3] => 3
[1,4,5,2,3] => [1,2,4,3,5] => 5
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00064: Permutations reversePermutations
St000054: 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] => 2
[2,1] => [1,2] => [2,1] => 2
[1,2,3] => [1,2,3] => [3,2,1] => 3
[1,3,2] => [1,2,3] => [3,2,1] => 3
[2,1,3] => [1,2,3] => [3,2,1] => 3
[2,3,1] => [1,2,3] => [3,2,1] => 3
[3,1,2] => [1,3,2] => [2,3,1] => 2
[3,2,1] => [1,3,2] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => 4
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => 4
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => 3
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => 3
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => 4
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => 4
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => 3
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => 3
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 4
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => 4
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => 4
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => 4
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => 2
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => 3
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => 2
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => 3
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => 3
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => 3
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => 5
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => 4
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 5
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => 3
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => 5
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => 3
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => 5
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
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00064: Permutations reversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000025: 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]
=> 2
[2,1] => [1,2] => [2,1] => [1,1,0,0]
=> 2
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 2
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[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]
=> 5
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
Description
The number of initial rises of a Dyck path. In other words, this is the height of the first peak of $D$.
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00066: Permutations inversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck 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] => [1,2] => [1,0,1,0]
=> 2
[2,1] => [1,2] => [1,2] => [1,0,1,0]
=> 2
[1,2,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[3,1,2] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,2,1] => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[3,1,4,2] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[3,2,4,1] => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[4,2,3,1] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[4,3,1,2] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[4,3,2,1] => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 5
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 5
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 $A$.
Matching statistic: St000007
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
Mp00252: Permutations restrictionPermutations
St000007: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [] => 0 = 1 - 1
[1,2] => [1,0,1,0]
=> [1,2] => [1] => 1 = 2 - 1
[2,1] => [1,1,0,0]
=> [2,1] => [1] => 1 = 2 - 1
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => [1,2] => 1 = 2 - 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,3,2] => [1,2] => 1 = 2 - 1
[2,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => [2,1] => 2 = 3 - 1
[2,3,1] => [1,1,0,1,0,0]
=> [2,3,1] => [2,1] => 2 = 3 - 1
[3,1,2] => [1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 2 = 3 - 1
[3,2,1] => [1,1,1,0,0,0]
=> [3,2,1] => [2,1] => 2 = 3 - 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3] => 1 = 2 - 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3] => 1 = 2 - 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2] => 2 = 3 - 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,2] => 2 = 3 - 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 2 = 3 - 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,2] => 2 = 3 - 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1] => 2 = 3 - 1
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,1] => 2 = 3 - 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 2 = 3 - 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,3,1] => 2 = 3 - 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 3 = 4 - 1
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 3 = 4 - 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1] => 3 = 4 - 1
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,1] => 3 = 4 - 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 3 = 4 - 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,2,1] => 3 = 4 - 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 3 = 4 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 3 = 4 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 3 = 4 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 3 = 4 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 3 = 4 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4] => 1 = 2 - 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4] => 1 = 2 - 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3] => 2 = 3 - 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,3] => 2 = 3 - 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,3] => 2 = 3 - 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,3,4,2] => 2 = 3 - 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,2] => 2 = 3 - 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,3,4,2] => 2 = 3 - 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 3 = 4 - 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 3 = 4 - 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2] => 3 = 4 - 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,4,3,2] => 3 = 4 - 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,4,3,2] => 3 = 4 - 1
Description
The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00088: Permutations Kreweras complementPermutations
Mp00061: Permutations to increasing treeBinary trees
St000051: Binary trees ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [.,.]
=> 0 = 1 - 1
[1,2] => [1,2] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
[2,1] => [1,2] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
[1,2,3] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[1,3,2] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[2,1,3] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[2,3,1] => [1,2,3] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[3,1,2] => [1,3,2] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,2,1] => [1,3,2] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[1,2,4,3] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[1,3,2,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[1,3,4,2] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[1,4,2,3] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[1,4,3,2] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[2,1,3,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[2,1,4,3] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[2,3,1,4] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [[.,[.,[.,.]]],.]
=> 3 = 4 - 1
[2,4,1,3] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[2,4,3,1] => [1,2,4,3] => [2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[3,1,2,4] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[3,1,4,2] => [1,3,4,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[3,2,4,1] => [1,3,4,2] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[3,4,2,1] => [1,3,2,4] => [2,4,3,1] => [[.,[[.,.],.]],.]
=> 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[4,2,1,3] => [1,4,3,2] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[4,3,1,2] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[4,3,2,1] => [1,4,2,3] => [2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> 4 = 5 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 4 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> 4 = 5 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> 4 = 5 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> 4 = 5 - 1
Description
The size of the left subtree of a binary tree.
Matching statistic: St000133
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00069: Permutations complementPermutations
Mp00088: Permutations Kreweras complementPermutations
St000133: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0 = 1 - 1
[1,2] => [1,2] => [2,1] => [1,2] => 1 = 2 - 1
[2,1] => [1,2] => [2,1] => [1,2] => 1 = 2 - 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,3,2] => 2 = 3 - 1
[3,1,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[3,2,1] => [1,3,2] => [3,1,2] => [3,1,2] => 1 = 2 - 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[1,4,2,3] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[1,4,3,2] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 3 = 4 - 1
[2,4,1,3] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[2,4,3,1] => [1,2,4,3] => [4,3,1,2] => [4,1,3,2] => 2 = 3 - 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[3,1,4,2] => [1,3,4,2] => [4,2,1,3] => [4,3,1,2] => 1 = 2 - 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[3,2,4,1] => [1,3,4,2] => [4,2,1,3] => [4,3,1,2] => 1 = 2 - 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,3,4,2] => 3 = 4 - 1
[4,1,2,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[4,2,1,3] => [1,4,3,2] => [4,1,2,3] => [3,4,1,2] => 1 = 2 - 1
[4,2,3,1] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[4,3,1,2] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[4,3,2,1] => [1,4,2,3] => [4,1,3,2] => [3,1,4,2] => 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,2,5,3,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,5,4,3,2] => 4 = 5 - 1
[1,3,5,2,4] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,4,3,1,2] => [5,1,4,3,2] => 3 = 4 - 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 4 = 5 - 1
[1,4,2,5,3] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,1,3,2] => 2 = 3 - 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 4 = 5 - 1
[1,4,3,5,2] => [1,2,4,5,3] => [5,4,2,1,3] => [5,4,1,3,2] => 2 = 3 - 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,4,2,3,1] => [1,4,5,3,2] => 4 = 5 - 1
Description
The "bounce" of a permutation.
Matching statistic: St000439
Mp00090: Permutations cycle-as-one-line notationPermutations
Mp00064: Permutations reversePermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
=> 2 = 1 + 1
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,3,2] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[2,1,3] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[2,3,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> 4 = 3 + 1
[3,1,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,3,2,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,3,4,2] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,4,2,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,4,3,2] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,1,3,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,1,4,3] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,3,1,4] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[2,4,1,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[3,1,2,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,1,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,2,1,4] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,2,4,1] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[3,4,2,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[4,1,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[4,2,1,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[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]
=> 6 = 5 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 5 = 4 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 6 = 5 + 1
Description
The position of the first down step of a Dyck path.
St000724: Permutations ⟶ ℤResult quality: 83% values known / values provided: 100%distinct values known / distinct values provided: 83%
Values
[1] => ? = 1
[1,2] => 2
[2,1] => 2
[1,2,3] => 3
[1,3,2] => 3
[2,1,3] => 2
[2,3,1] => 3
[3,1,2] => 2
[3,2,1] => 3
[1,2,3,4] => 4
[1,2,4,3] => 4
[1,3,2,4] => 3
[1,3,4,2] => 4
[1,4,2,3] => 3
[1,4,3,2] => 4
[2,1,3,4] => 2
[2,1,4,3] => 2
[2,3,1,4] => 3
[2,3,4,1] => 4
[2,4,1,3] => 4
[2,4,3,1] => 4
[3,1,2,4] => 4
[3,1,4,2] => 4
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 4
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 4
[4,2,3,1] => 3
[4,3,1,2] => 2
[4,3,2,1] => 4
[1,2,3,4,5] => 5
[1,2,3,5,4] => 5
[1,2,4,3,5] => 4
[1,2,4,5,3] => 5
[1,2,5,3,4] => 4
[1,2,5,4,3] => 5
[1,3,2,4,5] => 3
[1,3,2,5,4] => 3
[1,3,4,2,5] => 4
[1,3,4,5,2] => 5
[1,3,5,2,4] => 5
[1,3,5,4,2] => 5
[1,4,2,3,5] => 5
[1,4,2,5,3] => 5
[1,4,3,2,5] => 4
[1,4,3,5,2] => 4
[1,4,5,2,3] => 3
[1,4,5,3,2] => 5
Description
The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. Associate an increasing binary tree to the permutation using [[Mp00061]]. Then follow the path starting at the root which always selects the child with the smaller label. This statistic is the label of the leaf in the path, see [1]. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the greater neighbor of the maximum ([[St000060]]), see also [3].
St000060: Permutations ⟶ ℤResult quality: 83% values known / values provided: 100%distinct values known / distinct values provided: 83%
Values
[1] => ? = 1 - 1
[1,2] => 1 = 2 - 1
[2,1] => 1 = 2 - 1
[1,2,3] => 2 = 3 - 1
[1,3,2] => 2 = 3 - 1
[2,1,3] => 1 = 2 - 1
[2,3,1] => 2 = 3 - 1
[3,1,2] => 1 = 2 - 1
[3,2,1] => 2 = 3 - 1
[1,2,3,4] => 3 = 4 - 1
[1,2,4,3] => 3 = 4 - 1
[1,3,2,4] => 2 = 3 - 1
[1,3,4,2] => 3 = 4 - 1
[1,4,2,3] => 2 = 3 - 1
[1,4,3,2] => 3 = 4 - 1
[2,1,3,4] => 3 = 4 - 1
[2,1,4,3] => 3 = 4 - 1
[2,3,1,4] => 1 = 2 - 1
[2,3,4,1] => 3 = 4 - 1
[2,4,1,3] => 2 = 3 - 1
[2,4,3,1] => 3 = 4 - 1
[3,1,2,4] => 2 = 3 - 1
[3,1,4,2] => 2 = 3 - 1
[3,2,1,4] => 1 = 2 - 1
[3,2,4,1] => 2 = 3 - 1
[3,4,1,2] => 3 = 4 - 1
[3,4,2,1] => 3 = 4 - 1
[4,1,2,3] => 1 = 2 - 1
[4,1,3,2] => 1 = 2 - 1
[4,2,1,3] => 2 = 3 - 1
[4,2,3,1] => 2 = 3 - 1
[4,3,1,2] => 3 = 4 - 1
[4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => 4 = 5 - 1
[1,2,3,5,4] => 4 = 5 - 1
[1,2,4,3,5] => 3 = 4 - 1
[1,2,4,5,3] => 4 = 5 - 1
[1,2,5,3,4] => 3 = 4 - 1
[1,2,5,4,3] => 4 = 5 - 1
[1,3,2,4,5] => 4 = 5 - 1
[1,3,2,5,4] => 4 = 5 - 1
[1,3,4,2,5] => 2 = 3 - 1
[1,3,4,5,2] => 4 = 5 - 1
[1,3,5,2,4] => 3 = 4 - 1
[1,3,5,4,2] => 4 = 5 - 1
[1,4,2,3,5] => 3 = 4 - 1
[1,4,2,5,3] => 3 = 4 - 1
[1,4,3,2,5] => 2 = 3 - 1
[1,4,3,5,2] => 3 = 4 - 1
[1,4,5,2,3] => 4 = 5 - 1
[1,4,5,3,2] => 4 = 5 - 1
Description
The greater neighbor of the maximum. Han [2] showed that this statistic is (up to a shift) equidistributed on zigzag permutations (permutations $\pi$ such that $\pi(1) < \pi(2) > \pi(3) \cdots$) with the smallest path leaf label of the binary tree associated to a permutation ([[St000724]]), see also [3].
The following 51 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000800The number of occurrences of the vincular pattern |231 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000989The number of final rises of a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000454The largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000259The diameter of a connected graph. St001060The distinguishing index of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000200The row of the unique '1' in the last column of the alternating sign matrix. 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. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001557The number of inversions of the second entry of a permutation. St000264The girth of a graph, which is not a tree. St001330The hat guessing number of a graph. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St001668The number of points of the poset minus the width of the poset. St001580The acyclic chromatic number of a graph. St000272The treewidth of a graph. St000536The pathwidth of a graph. St000778The metric dimension of a graph. St001270The bandwidth of a graph. St001962The proper pathwidth of a graph. St001883The mutual visibility number of a graph. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000765The number of weak records in an integer composition. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001927Sparre Andersen's number of positives of a signed permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000822The Hadwiger number of the graph. St000299The number of nonisomorphic vertex-induced subtrees. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree.