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Your data matches 172 different statistics following compositions of up to 3 maps.
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Matching statistic: St000662
(load all 4 compositions to match this statistic)
(load all 4 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 9 compositions to match this statistic)
(load all 9 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 5 compositions to match this statistic)
(load all 5 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 5 compositions to match this statistic)
(load all 5 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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ 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] => [1,3,2] => 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] => [1,3,2] => 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] => [2,1,3] => 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] => [2,1,3] => 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] => [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] => [2,4,3,1] => 1
[2,1,1,1] => [2,3,4,1] => [1,4,3,2] => 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] => [2,4,3,1] => 1
[3,1,1,1] => [2,3,4,1] => [1,4,3,2] => 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] => [2,4,3,1] => 1
[4,1,1,1] => [2,3,4,1] => [1,4,3,2] => 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] => [3,2,4,1] => 1
[2,1,1,2] => [2,3,1,4] => [4,1,3,2] => 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 1
[2,2,1,1] => [3,4,1,2] => [2,1,4,3] => 2
[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] => [3,2,4,1] => 1
[1,3,1,2] => [1,3,4,2] => [2,4,3,1] => 1
[1,3,2,1] => [1,4,3,2] => [2,3,4,1] => 2
[2,1,1,3] => [2,3,1,4] => [4,1,3,2] => 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 1
[2,3,1,1] => [3,4,1,2] => [2,1,4,3] => 2
[3,1,1,2] => [2,3,4,1] => [1,4,3,2] => 1
[3,1,2,1] => [2,4,3,1] => [1,3,4,2] => 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: St000093
(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
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [2] => ([],2)
=> 2 = 1 + 1
[1,1,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,3] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,1,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,2,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,2,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,2,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[2,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[3,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [3] => ([],3)
=> 3 = 2 + 1
[1,1,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,1,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,3,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,4,1,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,1,1,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,2,2] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,2,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,2,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,2,3] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,3,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,1,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2,3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,3,2,1] => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[2,1,1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[2,3,1,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000325
(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
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[1,2,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[2,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[3,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,4,1] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,2] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,2,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[2,1,1,2] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,2,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,2,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,2] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,1,3] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,3,1] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,3,1,2] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,3,2,1] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[2,1,1,3] => [2,3,1,4] => [3,1,2,4] => 2 = 1 + 1
[2,1,3,1] => [2,4,1,3] => [3,1,4,2] => 3 = 2 + 1
[2,3,1,1] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[3,1,1,2] => [2,3,4,1] => [4,1,2,3] => 2 = 1 + 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 1
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000443
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(load all 9 compositions to match this statistic)
Mp00054: Parking functions —to inverse des composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1 = 0 + 1
[1,1] => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,1] => [1,0,1,0]
=> 1 = 0 + 1
[2,1] => [2] => [1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,3] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[3,1,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
[3,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,1] => [3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,4,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[4,1,1,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,2,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,3,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3,2,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[2,3,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
Description
The number of long tunnels of a Dyck path.
A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
The following 162 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. 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. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. 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. St001358The largest degree of a regular subgraph of a graph. 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. 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. St000172The Grundy number of a graph. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. 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. 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. 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. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. 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. St000439The position of the first down step of a Dyck path. 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. 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. 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. 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). 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001674The number of vertices of the largest induced star graph in the graph. St001812The biclique partition number of a graph. St001323The independence gap of a graph. 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. St000741The Colin de Verdière graph invariant. St001427The number of descents of a signed permutation. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001330The hat guessing number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000731The number of double exceedences of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000039The number of crossings of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St001726The number of visible inversions of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001935The number of ascents in a parking function. St000455The second largest eigenvalue of a graph if it is integral. St001896The number of right descents of a signed permutations. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001624The breadth of a lattice. St000761The number of ascents in an integer composition. St001777The number of weak descents in an integer composition. St001946The number of descents in a parking function. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000776The maximal multiplicity of an eigenvalue in a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001487The number of inner corners of a skew partition.
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