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Your data matches 277 different statistics following compositions of up to 3 maps.
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Matching statistic: St001935
St001935: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 1
[2,1] => 0
[1,1,1] => 0
[1,1,2] => 1
[1,2,1] => 1
[2,1,1] => 0
[1,1,3] => 1
[1,3,1] => 1
[3,1,1] => 0
[1,2,2] => 1
[2,1,2] => 1
[2,2,1] => 0
[1,2,3] => 2
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 0
[1,1,1,1] => 0
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,2,1,1] => 1
[2,1,1,1] => 0
[1,1,1,3] => 1
[1,1,3,1] => 1
[1,3,1,1] => 1
[3,1,1,1] => 0
[1,1,1,4] => 1
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 0
[1,1,2,2] => 1
[1,2,1,2] => 2
[1,2,2,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
[2,2,1,1] => 0
[1,1,2,3] => 2
[1,1,3,2] => 1
[1,2,1,3] => 2
[1,2,3,1] => 2
[1,3,1,2] => 2
[1,3,2,1] => 1
[2,1,1,3] => 1
[2,1,3,1] => 1
[2,3,1,1] => 1
[3,1,1,2] => 1
[3,1,2,1] => 1
Description
The number of ascents in a parking function.
This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St001946
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St001946: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 0
[1,2] => 0
[2,1] => 1
[1,1,1] => 0
[1,1,2] => 0
[1,2,1] => 1
[2,1,1] => 1
[1,1,3] => 0
[1,3,1] => 1
[3,1,1] => 1
[1,2,2] => 0
[2,1,2] => 1
[2,2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 2
[1,1,1,1] => 0
[1,1,1,2] => 0
[1,1,2,1] => 1
[1,2,1,1] => 1
[2,1,1,1] => 1
[1,1,1,3] => 0
[1,1,3,1] => 1
[1,3,1,1] => 1
[3,1,1,1] => 1
[1,1,1,4] => 0
[1,1,4,1] => 1
[1,4,1,1] => 1
[4,1,1,1] => 1
[1,1,2,2] => 0
[1,2,1,2] => 1
[1,2,2,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 2
[2,2,1,1] => 1
[1,1,2,3] => 0
[1,1,3,2] => 1
[1,2,1,3] => 1
[1,2,3,1] => 1
[1,3,1,2] => 1
[1,3,2,1] => 2
[2,1,1,3] => 1
[2,1,3,1] => 2
[2,3,1,1] => 1
[3,1,1,2] => 1
[3,1,2,1] => 2
Description
The number of descents in a parking function.
This is the number of indices $i$ such that $p_i > p_{i+1}$.
Matching statistic: St000662
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,1] => [1,2] => 0
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => 0
[1,1,2] => [1,2,3] => 0
[1,2,1] => [1,3,2] => 1
[2,1,1] => [2,3,1] => 1
[1,1,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => 1
[3,1,1] => [2,3,1] => 1
[1,2,2] => [1,2,3] => 0
[2,1,2] => [2,1,3] => 1
[2,2,1] => [3,1,2] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => 1
[3,1,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => 2
[1,1,1,1] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,4,3] => 1
[1,2,1,1] => [1,3,4,2] => 1
[2,1,1,1] => [2,3,4,1] => 1
[1,1,1,3] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,4,3] => 1
[1,3,1,1] => [1,3,4,2] => 1
[3,1,1,1] => [2,3,4,1] => 1
[1,1,1,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => 1
[1,4,1,1] => [1,3,4,2] => 1
[4,1,1,1] => [2,3,4,1] => 1
[1,1,2,2] => [1,2,3,4] => 0
[1,2,1,2] => [1,3,2,4] => 1
[1,2,2,1] => [1,4,2,3] => 1
[2,1,1,2] => [2,3,1,4] => 1
[2,1,2,1] => [2,4,1,3] => 1
[2,2,1,1] => [3,4,1,2] => 2
[1,1,2,3] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,4,3] => 1
[1,2,1,3] => [1,3,2,4] => 1
[1,2,3,1] => [1,4,2,3] => 1
[1,3,1,2] => [1,3,4,2] => 1
[1,3,2,1] => [1,4,3,2] => 2
[2,1,1,3] => [2,3,1,4] => 1
[2,1,3,1] => [2,4,1,3] => 1
[2,3,1,1] => [3,4,1,2] => 2
[3,1,1,2] => [2,3,4,1] => 1
[3,1,2,1] => [2,4,3,1] => 2
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St000021
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ 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] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,3,2] => [1,3,2] => 1
[2,1,1] => [2,3,1] => [3,1,2] => 1
[1,1,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,1] => [2,3,1] => [3,1,2] => 1
[1,2,2] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => 1
[2,2,1] => [3,1,2] => [2,3,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [2,3,1] => 1
[3,1,2] => [2,3,1] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 1
[2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 2
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 1
[1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 2
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000024
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,1] => [1,1] => [1,0,1,0]
=> 0
[1,2] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,1,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2,1] => [3] => [1,1,1,0,0,0]
=> 2
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[4,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,2,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,2,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000157
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ 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]]
=> 0
[1,2] => [1,2] => [[1,2]]
=> 0
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,1,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,1,2] => [1,2,3] => [[1,2,3]]
=> 0
[1,2,1] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,1] => [2,3,1] => [[1,3],[2]]
=> 1
[1,1,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,1] => [2,3,1] => [[1,3],[2]]
=> 1
[1,2,2] => [1,2,3] => [[1,2,3]]
=> 0
[2,1,2] => [2,1,3] => [[1,3],[2]]
=> 1
[2,2,1] => [3,1,2] => [[1,2],[3]]
=> 1
[1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 1
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1] => [3,1,2] => [[1,2],[3]]
=> 1
[3,1,2] => [2,3,1] => [[1,3],[2]]
=> 1
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,1,1,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,1,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,2,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,2,1,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[2,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,1,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,3,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,3,1,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[3,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,1,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,4,1,1] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[4,1,1,1] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[1,1,2,2] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,1,2] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,2,2,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[2,1,1,2] => [2,3,1,4] => [[1,3,4],[2]]
=> 1
[2,1,2,1] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[2,2,1,1] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,1,2,3] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,3,2] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,2,1,3] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,2,3,1] => [1,4,2,3] => [[1,2,3],[4]]
=> 1
[1,3,1,2] => [1,3,4,2] => [[1,2,4],[3]]
=> 1
[1,3,2,1] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[2,1,1,3] => [2,3,1,4] => [[1,3,4],[2]]
=> 1
[2,1,3,1] => [2,4,1,3] => [[1,3],[2,4]]
=> 2
[2,3,1,1] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[3,1,1,2] => [2,3,4,1] => [[1,3,4],[2]]
=> 1
[3,1,2,1] => [2,4,3,1] => [[1,3],[2],[4]]
=> 2
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000394
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,1] => [1,1] => [1,0,1,0]
=> 0
[1,2] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,1,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2,1] => [3] => [1,1,1,0,0,0]
=> 2
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[4,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,2,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,2,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St001189
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001189: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 0
[1,1] => [1,1] => [1,0,1,0]
=> 0
[1,2] => [1,1] => [1,0,1,0]
=> 0
[2,1] => [2] => [1,1,0,0]
=> 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,3] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[3,1,1] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,2,2] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[2,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2,1] => [3] => [1,1,1,0,0,0]
=> 2
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,4,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[4,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,2,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,2,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,3,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,3,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[2,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[3,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001298
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 0
[1,1] => [1,2] => [2,1] => 0
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[1,1,1] => [1,2,3] => [3,2,1] => 0
[1,1,2] => [1,2,3] => [3,2,1] => 0
[1,2,1] => [1,3,2] => [2,3,1] => 1
[2,1,1] => [2,3,1] => [2,1,3] => 1
[1,1,3] => [1,2,3] => [3,2,1] => 0
[1,3,1] => [1,3,2] => [2,3,1] => 1
[3,1,1] => [2,3,1] => [2,1,3] => 1
[1,2,2] => [1,2,3] => [3,2,1] => 0
[2,1,2] => [2,1,3] => [3,1,2] => 1
[2,2,1] => [3,1,2] => [1,3,2] => 1
[1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 1
[2,3,1] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [2,3,1] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 2
[1,1,1,1] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,1,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,2,1] => [1,2,4,3] => [3,4,2,1] => 1
[1,2,1,1] => [1,3,4,2] => [3,2,4,1] => 1
[2,1,1,1] => [2,3,4,1] => [3,2,1,4] => 1
[1,1,1,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,3,1] => [1,2,4,3] => [3,4,2,1] => 1
[1,3,1,1] => [1,3,4,2] => [3,2,4,1] => 1
[3,1,1,1] => [2,3,4,1] => [3,2,1,4] => 1
[1,1,1,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,4,1] => [1,2,4,3] => [3,4,2,1] => 1
[1,4,1,1] => [1,3,4,2] => [3,2,4,1] => 1
[4,1,1,1] => [2,3,4,1] => [3,2,1,4] => 1
[1,1,2,2] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,1,2] => [1,3,2,4] => [4,2,3,1] => 1
[1,2,2,1] => [1,4,2,3] => [2,4,3,1] => 1
[2,1,1,2] => [2,3,1,4] => [4,2,1,3] => 1
[2,1,2,1] => [2,4,1,3] => [2,1,4,3] => 2
[2,2,1,1] => [3,4,1,2] => [3,1,4,2] => 1
[1,1,2,3] => [1,2,3,4] => [4,3,2,1] => 0
[1,1,3,2] => [1,2,4,3] => [3,4,2,1] => 1
[1,2,1,3] => [1,3,2,4] => [4,2,3,1] => 1
[1,2,3,1] => [1,4,2,3] => [2,4,3,1] => 1
[1,3,1,2] => [1,3,4,2] => [3,2,4,1] => 1
[1,3,2,1] => [1,4,3,2] => [2,3,4,1] => 2
[2,1,1,3] => [2,3,1,4] => [4,2,1,3] => 1
[2,1,3,1] => [2,4,1,3] => [2,1,4,3] => 2
[2,3,1,1] => [3,4,1,2] => [3,1,4,2] => 1
[3,1,1,2] => [2,3,4,1] => [3,2,1,4] => 1
[3,1,2,1] => [2,4,3,1] => [2,1,3,4] => 2
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St001896
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001896: Signed permutations ⟶ 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] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,1,1] => [1,2,3] => [1,2,3] => 0
[1,1,2] => [1,2,3] => [1,2,3] => 0
[1,2,1] => [1,3,2] => [1,3,2] => 1
[2,1,1] => [2,3,1] => [2,3,1] => 1
[1,1,3] => [1,2,3] => [1,2,3] => 0
[1,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,1] => [2,3,1] => [2,3,1] => 1
[1,2,2] => [1,2,3] => [1,2,3] => 0
[2,1,2] => [2,1,3] => [2,1,3] => 1
[2,2,1] => [3,1,2] => [3,1,2] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 1
[3,1,2] => [2,3,1] => [2,3,1] => 1
[3,2,1] => [3,2,1] => [3,2,1] => 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,1] => [1,3,4,2] => [1,3,4,2] => 1
[2,1,1,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,1,1] => [1,3,4,2] => [1,3,4,2] => 1
[3,1,1,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,4,1,1] => [1,3,4,2] => [1,3,4,2] => 1
[4,1,1,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,2,1] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,1,2] => [2,3,1,4] => [2,3,1,4] => 1
[2,1,2,1] => [2,4,1,3] => [2,4,1,3] => 2
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[1,2,3,1] => [1,4,2,3] => [1,4,2,3] => 1
[1,3,1,2] => [1,3,4,2] => [1,3,4,2] => 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 2
[2,1,1,3] => [2,3,1,4] => [2,3,1,4] => 1
[2,1,3,1] => [2,4,1,3] => [2,4,1,3] => 2
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 1
[3,1,1,2] => [2,3,4,1] => [2,3,4,1] => 1
[3,1,2,1] => [2,4,3,1] => [2,4,3,1] => 2
Description
The number of right descents of a signed permutations.
An index is a right descent if it is a left descent of the inverse signed permutation.
The following 267 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000093The cardinality of a maximal independent set of vertices of a graph. St000172The Grundy number of a graph. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001389The number of partitions of the same length below the given integer partition. St001581The achromatic number of a graph. 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. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000039The number of crossings of a permutation. St000053The number of valleys of the Dyck path. St000067The inversion number of the alternating sign matrix. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000288The number of ones in a binary word. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000441The number of successions of a permutation. St000536The pathwidth of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000731The number of double exceedences of a permutation. St000877The depth of the binary word interpreted as a path. St000996The number of exclusive left-to-right maxima of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. St001176The size of a partition minus its first part. 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. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001323The independence gap of a graph. St001358The largest degree of a regular subgraph of a graph. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. 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. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001812The biclique partition number of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000507The number of ascents of a standard tableau. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama 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:
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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001494The Alon-Tarsi number of a graph. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001726The number of visible inversions of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St001963The tree-depth of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000439The position of the first down step of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. 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. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001674The number of vertices of the largest induced star graph in the graph. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000354The number of recoils of a permutation. St001480The number of simple summands of the module J^2/J^3. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001372The length of a longest cyclic run of ones of a binary word. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000702The number of weak deficiencies of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000761The number of ascents in an integer composition. St001777The number of weak descents in an integer composition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001571The Cartan determinant of the integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000937The number of positive values of the symmetric group character corresponding to the partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001498The normalised height of a Nakayama algebra with magnitude 1. St000260The radius of a connected graph. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000741The Colin de Verdière graph invariant. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001378The product of the cohook lengths of the integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001527The cyclic permutation representation number of an integer partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001933The largest multiplicity of a part in an integer partition. St001624The breadth of a lattice. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000177The number of free tiles in the pattern. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000455The second largest eigenvalue of a graph if it is integral. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000776The maximal multiplicity of an eigenvalue in a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000567The sum of the products of all pairs of parts. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St000422The energy of a graph, if it is integral. 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. St000456The monochromatic index of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001487The number of inner corners of a skew partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
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