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Your data matches 232 different statistics following compositions of up to 3 maps.
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Matching statistic: St001291
(load all 172 compositions to match this statistic)
(load all 172 compositions to match this statistic)
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 3
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 4
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> 2
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: St000054
(load all 61 compositions to match this statistic)
(load all 61 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ 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
[1,0]
=> [1] => 1
[1,0,1,0]
=> [2,1] => 2
[1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [3,2,1] => 3
[1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [3,1,2] => 3
[1,1,0,1,0,0]
=> [2,1,3] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 2
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: St001497
(load all 70 compositions to match this statistic)
(load all 70 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001497: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001497: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1
[1,0,1,0]
=> [1,2] => 2
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 3
[1,1,0,1,0,0]
=> [2,3,1] => 2
[1,1,1,0,0,0]
=> [3,1,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 3
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 4
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 4
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 4
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 3
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 5
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 5
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 4
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 5
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 4
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 5
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 5
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 2
Description
The position of the largest weak excedence of a permutation.
Matching statistic: St000141
(load all 51 compositions to match this statistic)
(load all 51 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000147
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> 0 = 1 - 1
[1,0,1,0]
=> [1]
=> 1 = 2 - 1
[1,1,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [2]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> []
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 2 - 1
Description
The largest part of an integer partition.
Matching statistic: St000316
(load all 93 compositions to match this statistic)
(load all 93 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0 = 1 - 1
[1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [3,1,2] => 2 = 3 - 1
[1,1,0,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
Description
The number of non-left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a **non-left-to-right-maximum** if there exists a $j < i$ such that $\sigma_j > \sigma_i$.
Matching statistic: St001227
(load all 55 compositions to match this statistic)
(load all 55 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000013
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: 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
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000025
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ 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
[1,0]
=> [1] => [1,0]
=> 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
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: St000026
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [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]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,0,0,1,0,0]
=> [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]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [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]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 5
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 5
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
Description
The position of the first return of a Dyck path.
The following 222 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000066The column of the unique '1' in the first row of the alternating sign matrix. St000240The number of indices that are not small excedances. St000443The number of long tunnels of a Dyck path. St000734The last entry in the first row of a standard tableau. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000839The largest opener of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001809The index of the step at the first peak of maximal height in a Dyck path. St000010The length of the partition. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000204The number of internal nodes of a binary tree. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000166The depth minus 1 of an ordered tree. St000297The number of leading ones in a binary word. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000505The biggest entry in the block containing the 1. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000730The maximal arc length of a set partition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000874The position of the last double rise in a Dyck path. St000971The smallest closer of a set partition. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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 special CNakayama 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:
St001268The size of the largest ordinal summand in the poset. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001725The harmonious chromatic number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001778The largest greatest common divisor of an element and its image in a permutation. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000053The number of valleys of the Dyck path. St000094The depth of an ordered tree. St000120The number of left tunnels of a Dyck path. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000171The degree of the graph. St000209Maximum difference of elements in cycles. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000356The number of occurrences of the pattern 13-2. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000497The lcb statistic of a set partition. St000521The number of distinct subtrees of an ordered tree. St000572The dimension exponent of a set partition. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001117The game chromatic index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001489The maximum of the number of descents and the number of inverse descents. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001841The number of inversions of a set partition. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St000061The number of nodes on the left branch of a binary tree. St000485The length of the longest cycle of a permutation. St000675The number of centered multitunnels of a Dyck path. St000692Babson and Steingrímsson's statistic of a permutation. St000702The number of weak deficiencies of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000503The maximal difference between two elements in a common block. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000728The dimension of a set partition. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000956The maximal displacement of a permutation. St000989The number of final rises of a permutation. St001077The prefix exchange distance of a permutation. St001721The degree of a binary word. St000784The maximum of the length and the largest part of the integer partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000733The row containing the largest entry of a standard tableau. St001118The acyclic chromatic index of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000691The number of changes of a binary word. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000326The position of the first one in a binary word after appending a 1 at the end. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000225Difference between largest and smallest parts in a partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000668The least common multiple of the parts of the partition. St000143The largest repeated part of a partition. St000840The number of closers smaller than the largest opener in a perfect matching. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001726The number of visible inversions of a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. 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. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000746The number of pairs with odd minimum in a perfect matching. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St000028The number of stack-sorts needed to sort a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000314The number of left-to-right-maxima of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000833The comajor index of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000060The greater neighbor of the maximum. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St001589The nesting number of a perfect matching. St000005The bounce statistic of a Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001584The area statistic between a Dyck path and its bounce path. St000327The number of cover relations in a poset. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001861The number of Bruhat lower covers of a permutation. St000041The number of nestings of a perfect matching. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001769The reflection length of a signed permutation. St000665The number of rafts of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001557The number of inversions of the second entry of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001877Number of indecomposable injective modules with projective dimension 2.
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