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Your data matches 110 different statistics following compositions of up to 3 maps.
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Matching statistic: St000245
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00066: Permutations —inverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 0
[1,2] => [1,2] => 1
[2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => 2
[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] => 0
[1,2,3,4] => [1,2,3,4] => 3
[1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,4,2,3] => 2
[1,4,2,3] => [1,3,4,2] => 2
[1,4,3,2] => [1,4,3,2] => 1
[2,1,3,4] => [2,1,3,4] => 2
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,1,2,4] => 2
[2,3,4,1] => [4,1,2,3] => 2
[2,4,1,3] => [3,1,4,2] => 1
[2,4,3,1] => [4,1,3,2] => 1
[3,1,2,4] => [2,3,1,4] => 2
[3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => 1
[3,2,4,1] => [4,2,1,3] => 1
[3,4,1,2] => [3,4,1,2] => 2
[3,4,2,1] => [4,3,1,2] => 1
[4,1,2,3] => [2,3,4,1] => 2
[4,1,3,2] => [2,4,3,1] => 1
[4,2,1,3] => [3,2,4,1] => 1
[4,2,3,1] => [4,2,3,1] => 1
[4,3,1,2] => [3,4,2,1] => 1
[4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => 4
[1,2,3,5,4] => [1,2,3,5,4] => 3
[1,2,4,3,5] => [1,2,4,3,5] => 3
[1,2,4,5,3] => [1,2,5,3,4] => 3
[1,2,5,3,4] => [1,2,4,5,3] => 3
[1,2,5,4,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [1,3,2,4,5] => 3
[1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,4,2,3,5] => 3
[1,3,4,5,2] => [1,5,2,3,4] => 3
[1,3,5,2,4] => [1,4,2,5,3] => 2
[1,3,5,4,2] => [1,5,2,4,3] => 2
[1,4,2,3,5] => [1,3,4,2,5] => 3
[1,4,2,5,3] => [1,3,5,2,4] => 3
[1,4,3,2,5] => [1,4,3,2,5] => 2
[1,4,3,5,2] => [1,5,3,2,4] => 2
[1,4,5,2,3] => [1,4,5,2,3] => 3
Description
The number of ascents of a permutation.
Matching statistic: St000021
(load all 45 compositions to match this statistic)
(load all 45 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ 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
[1,2] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => 2
[1,3,2] => [2,3,1] => [3,1,2] => 1
[2,1,3] => [3,1,2] => [2,3,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => 2
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => 2
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => 2
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => 1
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => 2
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 2
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => 1
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => 2
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => 1
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => 2
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => 1
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => 3
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => 3
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => 3
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => 3
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => 3
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => 3
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => 3
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => 3
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: St000157
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00084: Standard tableaux —conjugate⟶ 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
[1,2] => [[1,2]]
=> [[1],[2]]
=> 1
[2,1] => [[1],[2]]
=> [[1,2]]
=> 0
[1,2,3] => [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[2,1,3] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,3,1] => [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3,1,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[3,2,1] => [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[1,2,3,4] => [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3
[1,2,4,3] => [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
[1,3,2,4] => [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2
[1,3,4,2] => [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2
[1,4,2,3] => [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[2,3,1,4] => [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[2,3,4,1] => [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[2,4,1,3] => [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 1
[2,4,3,1] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[3,1,2,4] => [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 2
[3,1,4,2] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,2,4,1] => [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,4,1,2] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,4,2,1] => [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[4,1,2,3] => [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2
[4,1,3,2] => [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[4,2,1,3] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[4,3,1,2] => [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3
[1,2,4,5,3] => [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3
[1,2,5,3,4] => [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 3
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3
[1,3,4,5,2] => [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 3
[1,3,5,2,4] => [[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 2
[1,3,5,4,2] => [[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 3
[1,4,2,5,3] => [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 3
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: St000662
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,3,2] => [2,3,1] => [3,1,2] => [1,3,2] => 1
[2,1,3] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,3,1] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [1,4,3,2] => 2
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [4,1,3,2] => 2
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [2,4,3,1] => 2
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [3,1,4,2] => 2
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 1
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [4,3,1,2] => 2
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [1,4,2,3] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [4,2,3,1] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [3,4,2,1] => 2
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [1,3,4,2] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [2,1,4,3] => 1
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [4,2,1,3] => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 1
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [2,4,1,3] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [3,2,4,1] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [2,3,4,1] => 1
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [2,3,1,4] => 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [1,5,4,3,2] => 3
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [5,1,4,3,2] => 3
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [2,5,4,3,1] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [4,1,5,3,2] => 3
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => [1,2,5,4,3] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [5,4,1,3,2] => 3
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => [1,5,2,4,3] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [5,2,4,3,1] => 3
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [3,5,4,2,1] => 3
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [1,3,5,4,2] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => [2,1,5,4,3] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [5,3,1,4,2] => 3
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [4,5,1,3,2] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [5,1,2,4,3] => 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => [2,5,1,4,3] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [4,2,5,3,1] => 3
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: St000703
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [2,1] => [2,1] => [2,1] => 1
[2,1] => [1,2] => [1,2] => [1,2] => 0
[1,2,3] => [3,2,1] => [3,2,1] => [3,1,2] => 2
[1,3,2] => [2,3,1] => [3,1,2] => [2,3,1] => 1
[2,1,3] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1
[3,1,2] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[3,2,1] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 3
[1,2,4,3] => [3,4,2,1] => [4,3,1,2] => [2,4,1,3] => 2
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [4,3,1,2] => 2
[1,3,4,2] => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
[1,4,2,3] => [3,2,4,1] => [4,2,1,3] => [3,1,4,2] => 2
[1,4,3,2] => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 1
[2,1,3,4] => [4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 2
[2,1,4,3] => [3,4,1,2] => [3,4,1,2] => [2,4,3,1] => 1
[2,3,1,4] => [4,1,3,2] => [2,4,3,1] => [4,2,1,3] => 2
[2,3,4,1] => [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [3,2,4,1] => 1
[2,4,3,1] => [1,3,4,2] => [1,4,2,3] => [1,3,4,2] => 1
[3,1,2,4] => [4,2,1,3] => [3,2,4,1] => [4,3,2,1] => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[3,2,1,4] => [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 1
[3,2,4,1] => [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 1
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[3,4,2,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 2
[4,1,3,2] => [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 1
[4,2,1,3] => [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[4,3,1,2] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [5,1,2,3,4] => 4
[1,2,3,5,4] => [4,5,3,2,1] => [5,4,3,1,2] => [2,5,1,3,4] => 3
[1,2,4,3,5] => [5,3,4,2,1] => [5,4,2,3,1] => [5,3,1,2,4] => 3
[1,2,4,5,3] => [3,5,4,2,1] => [5,4,1,3,2] => [3,5,2,1,4] => 3
[1,2,5,3,4] => [4,3,5,2,1] => [5,4,2,1,3] => [3,1,5,2,4] => 3
[1,2,5,4,3] => [3,4,5,2,1] => [5,4,1,2,3] => [2,3,5,1,4] => 2
[1,3,2,4,5] => [5,4,2,3,1] => [5,3,4,2,1] => [5,1,4,2,3] => 3
[1,3,2,5,4] => [4,5,2,3,1] => [5,3,4,1,2] => [2,5,4,1,3] => 2
[1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => [5,4,1,3,2] => 3
[1,3,4,5,2] => [2,5,4,3,1] => [5,1,4,3,2] => [4,5,2,3,1] => 3
[1,3,5,2,4] => [4,2,5,3,1] => [5,2,4,1,3] => [3,4,5,1,2] => 2
[1,3,5,4,2] => [2,4,5,3,1] => [5,1,4,2,3] => [4,3,5,2,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [5,4,2,1,3] => 3
[1,4,2,5,3] => [3,5,2,4,1] => [5,3,1,4,2] => [4,5,1,2,3] => 3
[1,4,3,2,5] => [5,2,3,4,1] => [5,2,3,4,1] => [5,3,4,1,2] => 2
[1,4,3,5,2] => [2,5,3,4,1] => [5,1,3,4,2] => [3,5,4,2,1] => 2
[1,4,5,2,3] => [3,2,5,4,1] => [5,2,1,4,3] => [4,1,5,3,2] => 3
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000024
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 1
[2,1] => [[.,.],.]
=> [1,0,1,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[] => .
=> ?
=> ? = 0
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: St000053
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[] => .
=> ?
=> ? = 0
Description
The number of valleys of the Dyck path.
Matching statistic: St000168
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000168: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [[]]
=> 0
[1,2] => [.,[.,.]]
=> [[[]]]
=> 1
[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
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> 3
[] => .
=> ?
=> ? = 0
Description
The number of internal nodes of an ordered tree.
A node is internal if it is neither the root nor a leaf.
Matching statistic: St000316
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000316: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1] => 0
[1,2] => [.,[.,.]]
=> [2,1] => 1
[2,1] => [[.,.],.]
=> [1,2] => 0
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => 2
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => 1
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 3
[] => .
=> ? => ? = 0
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: St001169
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 0
[1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 0
[1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 2
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> 1
[3,1,2] => [[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> 1
[3,2,1] => [[[.,.],.],.]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> 2
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> 1
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[] => .
=> ?
=> ? = 0
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
The following 100 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. 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. St000542The number of left-to-right-minima of a permutation. St000676The number of odd rises of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000010The length of the partition. St000052The number of valleys of a Dyck path not on the x-axis. St000120The number of left tunnels of a Dyck path. St000141The maximum drop size of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000292The number of ascents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000632The jump number of the poset. St000672The number of minimal elements in Bruhat order not less than the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001176The size of a partition minus its first part. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000031The number of cycles in the cycle decomposition of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000105The number of blocks in the set partition. St000147The largest part of an integer partition. St000164The number of short pairs. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000390The number of runs of ones in a binary word. St000507The number of ascents of a standard tableau. St000527The width of the poset. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001304The number of maximally independent sets of vertices of a graph. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001581The achromatic number of a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St001480The number of simple summands of the module J^2/J^3. St000061The number of nodes on the left branch of a binary tree. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000925The number of topologically connected components of a set partition. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001812The biclique partition number of a graph. St000159The number of distinct parts of the integer partition. St000746The number of pairs with odd minimum in a perfect matching. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001935The number of ascents in a parking function. St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. St001668The number of points of the poset minus the width of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. 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 $eA$.
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