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Matching statistic: St000662
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000662: 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] => 0
[2,1] => [2,1] => [1] => 0
[1,2,3] => [1,2,3] => [1,2] => 0
[1,3,2] => [2,3,1] => [2,1] => 1
[2,1,3] => [2,1,3] => [2,1] => 1
[2,3,1] => [3,1,2] => [1,2] => 0
[3,1,2] => [1,3,2] => [1,2] => 0
[3,2,1] => [3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [2,3,4,1] => [2,3,1] => 1
[1,3,2,4] => [2,3,1,4] => [2,3,1] => 1
[1,3,4,2] => [2,4,1,3] => [2,1,3] => 1
[1,4,2,3] => [2,1,4,3] => [2,1,3] => 1
[1,4,3,2] => [3,4,2,1] => [3,2,1] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [3,2,4,1] => [3,2,1] => 2
[2,3,1,4] => [3,1,2,4] => [3,1,2] => 1
[2,3,4,1] => [4,1,2,3] => [1,2,3] => 0
[2,4,1,3] => [1,3,4,2] => [1,3,2] => 1
[2,4,3,1] => [4,2,3,1] => [2,3,1] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2] => 1
[3,1,4,2] => [3,4,1,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1] => 2
[3,2,4,1] => [4,2,1,3] => [2,1,3] => 1
[3,4,1,2] => [1,4,2,3] => [1,2,3] => 0
[3,4,2,1] => [4,3,1,2] => [3,1,2] => 1
[4,1,2,3] => [1,2,4,3] => [1,2,3] => 0
[4,1,3,2] => [2,4,3,1] => [2,3,1] => 1
[4,2,1,3] => [3,1,4,2] => [3,1,2] => 1
[4,2,3,1] => [4,1,3,2] => [1,3,2] => 1
[4,3,1,2] => [1,4,3,2] => [1,3,2] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,1] => 1
[1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1] => 1
[1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => 1
[1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,4] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2
[1,3,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => 1
[1,3,2,5,4] => [3,4,2,5,1] => [3,4,2,1] => 2
[1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3] => 1
[1,3,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => 1
[1,3,5,2,4] => [2,1,4,5,3] => [2,1,4,3] => 1
[1,3,5,4,2] => [3,5,2,4,1] => [3,2,4,1] => 2
[1,4,2,3,5] => [2,1,4,3,5] => [2,1,4,3] => 1
[1,4,2,5,3] => [3,4,5,1,2] => [3,4,1,2] => 2
[1,4,3,2,5] => [3,4,2,1,5] => [3,4,2,1] => 2
[1,4,3,5,2] => [3,5,2,1,4] => [3,2,1,4] => 2
[1,4,5,2,3] => [2,1,5,3,4] => [2,1,3,4] => 1
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: St000470
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [] => [] => ? = 0 + 1
[1,2] => [1,2] => [1] => [1] => 1 = 0 + 1
[2,1] => [2,1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2] => [1,2] => 1 = 0 + 1
[1,3,2] => [2,3,1] => [2,1] => [2,1] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [2,3,4,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,3,2,4] => [2,3,1,4] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,3,4,2] => [2,4,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,4,2,3] => [2,1,4,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,4,3,2] => [3,4,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,1,4,3] => [3,2,4,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,4,1,3] => [1,3,4,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[2,4,3,1] => [4,2,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,4,1,2] => [1,4,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,4,2,1] => [4,3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,1,2,3] => [1,2,4,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [2,4,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[4,2,1,3] => [3,1,4,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,2,3,1] => [4,1,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,3,1,2] => [1,4,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,3,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,2,5,4] => [3,4,2,5,1] => [3,4,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[1,3,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,3,5,2,4] => [2,1,4,5,3] => [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,3,5,4,2] => [3,5,2,4,1] => [3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[1,4,2,3,5] => [2,1,4,3,5] => [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,2,5,3] => [3,4,5,1,2] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,4,3,2,5] => [3,4,2,1,5] => [3,4,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,4,3,5,2] => [3,5,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[1,4,5,2,3] => [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,4,5,3,2] => [3,5,4,1,2] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5 + 1
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 4 + 1
[5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3 + 1
[4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2 + 1
[4,5,6,7,3,8,2,1] => [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 2 + 1
[4,5,6,3,7,8,2,1] => [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => ? = 2 + 1
[5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => ? = 3 + 1
[3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1 + 1
[4,5,6,7,8,2,3,1] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => [3,4,5,6,7,1,2] => ? = 1 + 1
[6,7,8,5,2,3,4,1] => [8,1,2,7,6,3,4,5] => [1,2,7,6,3,4,5] => [5,6,7,4,1,2,3] => ? = 2 + 1
[8,5,6,7,2,3,4,1] => [8,1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [7,4,5,6,1,2,3] => ? = 2 + 1
[5,6,7,8,2,3,4,1] => [8,1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1 + 1
[8,7,6,2,3,4,5,1] => [8,1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3 + 1
[6,7,8,2,3,4,5,1] => [8,1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1 + 1
[8,7,2,3,4,5,6,1] => [8,1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 2 + 1
[7,8,2,3,4,5,6,1] => [8,1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1 + 1
[8,2,3,4,5,6,7,1] => [8,1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 5 + 1
[7,6,5,2,3,4,8,1] => [8,1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 3 + 1
[7,6,2,3,4,5,8,1] => [8,1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 2 + 1
[5,4,3,2,6,7,8,1] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3 + 1
[3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 1 + 1
[5,4,2,3,6,7,8,1] => [8,1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2 + 1
[4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 2 + 1
[3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 1 + 1
[3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1 + 1
[8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 5 + 1
[7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [6,7,4,5,1,2,3] => ? = 2 + 1
[7,5,6,8,3,4,1,2] => [1,8,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [6,4,5,7,1,2,3] => ? = 2 + 1
[5,6,7,8,3,4,1,2] => [1,8,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1 + 1
[7,8,5,3,4,6,1,2] => [1,8,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => ? = 2 + 1
[7,8,3,4,5,6,1,2] => [1,8,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1 + 1
[8,3,4,5,6,7,1,2] => [1,8,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[7,5,6,3,4,8,1,2] => [1,8,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [6,4,5,1,2,3,7] => ? = 2 + 1
[5,6,7,3,4,8,1,2] => [1,8,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => ? = 1 + 1
[7,5,3,4,6,8,1,2] => [1,8,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => ? = 2 + 1
[7,3,4,5,6,8,1,2] => [1,8,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1 + 1
[5,6,3,4,7,8,1,2] => [1,8,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => ? = 1 + 1
[6,4,3,5,7,8,1,2] => [1,8,4,2,5,3,6,7] => [1,4,2,5,3,6,7] => [5,3,1,2,4,6,7] => ? = 2 + 1
[5,3,4,6,7,8,1,2] => [1,8,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1 + 1
[4,3,5,6,7,8,1,2] => [1,8,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1 + 1
[4,5,6,7,8,2,1,3] => [7,1,8,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1 + 1
[6,7,8,5,4,1,2,3] => [1,2,8,7,6,3,4,5] => [1,2,7,6,3,4,5] => [5,6,7,4,1,2,3] => ? = 2 + 1
[8,6,5,7,4,1,2,3] => [1,2,8,7,5,3,6,4] => [1,2,7,5,3,6,4] => [7,5,4,6,1,2,3] => ? = 3 + 1
[5,6,7,8,4,1,2,3] => [1,2,8,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1 + 1
[6,7,8,4,5,1,2,3] => [1,2,8,3,7,4,5,6] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1 + 1
[8,7,5,4,6,1,2,3] => [1,2,8,6,3,7,5,4] => [1,2,6,3,7,5,4] => [7,6,4,1,2,3,5] => ? = 3 + 1
[6,5,4,7,8,1,2,3] => [1,2,8,5,4,3,6,7] => [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => ? = 2 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000021
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 87%●distinct values known / distinct values provided: 60%
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 87%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [] => [] => 0
[1,2] => [1,2] => [1] => [1] => 0
[2,1] => [2,1] => [1] => [1] => 0
[1,2,3] => [1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [2,3,1] => [2,1] => [2,1] => 1
[2,1,3] => [2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [3,1,2] => [1,2] => [1,2] => 0
[3,1,2] => [1,3,2] => [1,2] => [1,2] => 0
[3,2,1] => [3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [2,3,4,1] => [2,3,1] => [1,3,2] => 1
[1,3,2,4] => [2,3,1,4] => [2,3,1] => [1,3,2] => 1
[1,3,4,2] => [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[1,4,2,3] => [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,4,3,2] => [3,4,2,1] => [3,2,1] => [3,2,1] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [3,2,4,1] => [3,2,1] => [3,2,1] => 2
[2,3,1,4] => [3,1,2,4] => [3,1,2] => [2,3,1] => 1
[2,3,4,1] => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[2,4,1,3] => [1,3,4,2] => [1,3,2] => [3,1,2] => 1
[2,4,3,1] => [4,2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2,4] => [1,3,2,4] => [1,3,2] => [3,1,2] => 1
[3,1,4,2] => [3,4,1,2] => [3,1,2] => [2,3,1] => 1
[3,2,1,4] => [3,2,1,4] => [3,2,1] => [3,2,1] => 2
[3,2,4,1] => [4,2,1,3] => [2,1,3] => [2,1,3] => 1
[3,4,1,2] => [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[3,4,2,1] => [4,3,1,2] => [3,1,2] => [2,3,1] => 1
[4,1,2,3] => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [2,4,3,1] => [2,3,1] => [1,3,2] => 1
[4,2,1,3] => [3,1,4,2] => [3,1,2] => [2,3,1] => 1
[4,2,3,1] => [4,1,3,2] => [1,3,2] => [3,1,2] => 1
[4,3,1,2] => [1,4,3,2] => [1,3,2] => [3,1,2] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,1] => [1,2,4,3] => 1
[1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1] => [1,2,4,3] => 1
[1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => [1,3,2,4] => 1
[1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,4] => [1,3,2,4] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [1,4,3,2] => 2
[1,3,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => [1,3,2,4] => 1
[1,3,2,5,4] => [3,4,2,5,1] => [3,4,2,1] => [1,4,3,2] => 2
[1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3] => [1,3,4,2] => 1
[1,3,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,3,5,2,4] => [2,1,4,5,3] => [2,1,4,3] => [1,4,2,3] => 1
[1,3,5,4,2] => [3,5,2,4,1] => [3,2,4,1] => [2,1,4,3] => 2
[1,4,2,3,5] => [2,1,4,3,5] => [2,1,4,3] => [1,4,2,3] => 1
[1,4,2,5,3] => [3,4,5,1,2] => [3,4,1,2] => [3,1,4,2] => 2
[1,4,3,2,5] => [3,4,2,1,5] => [3,4,2,1] => [1,4,3,2] => 2
[1,4,3,5,2] => [3,5,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,4,5,2,3] => [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 4
[5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3
[4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2
[4,5,6,7,3,8,2,1] => [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 2
[4,5,6,3,7,8,2,1] => [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => ? = 2
[5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => ? = 3
[3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1
[4,5,6,7,8,2,3,1] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => [3,4,5,6,7,1,2] => ? = 1
[6,7,8,5,2,3,4,1] => [8,1,2,7,6,3,4,5] => [1,2,7,6,3,4,5] => [5,6,7,4,1,2,3] => ? = 2
[8,5,6,7,2,3,4,1] => [8,1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [7,4,5,6,1,2,3] => ? = 2
[5,6,7,8,2,3,4,1] => [8,1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1
[8,7,6,2,3,4,5,1] => [8,1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3
[6,7,8,2,3,4,5,1] => [8,1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[8,7,2,3,4,5,6,1] => [8,1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 2
[7,8,2,3,4,5,6,1] => [8,1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[8,2,3,4,5,6,7,1] => [8,1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 5
[7,6,5,2,3,4,8,1] => [8,1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 3
[7,6,2,3,4,5,8,1] => [8,1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 2
[5,4,3,2,6,7,8,1] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3
[3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 1
[5,4,2,3,6,7,8,1] => [8,1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2
[4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 2
[3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 1
[3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1
[2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 5
[7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [6,7,4,5,1,2,3] => ? = 2
[7,5,6,8,3,4,1,2] => [1,8,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [6,4,5,7,1,2,3] => ? = 2
[5,6,7,8,3,4,1,2] => [1,8,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1
[7,8,5,3,4,6,1,2] => [1,8,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => ? = 2
[7,8,3,4,5,6,1,2] => [1,8,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1
[8,3,4,5,6,7,1,2] => [1,8,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[7,5,6,3,4,8,1,2] => [1,8,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [6,4,5,1,2,3,7] => ? = 2
[5,6,7,3,4,8,1,2] => [1,8,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => ? = 1
[7,5,3,4,6,8,1,2] => [1,8,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => ? = 2
[7,3,4,5,6,8,1,2] => [1,8,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[5,6,3,4,7,8,1,2] => [1,8,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => ? = 1
[6,4,3,5,7,8,1,2] => [1,8,4,2,5,3,6,7] => [1,4,2,5,3,6,7] => [5,3,1,2,4,6,7] => ? = 2
[5,3,4,6,7,8,1,2] => [1,8,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[4,3,5,6,7,8,1,2] => [1,8,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[3,4,5,6,7,8,1,2] => [1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[4,5,6,7,8,2,1,3] => [7,1,8,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1
[6,7,8,5,4,1,2,3] => [1,2,8,7,6,3,4,5] => [1,2,7,6,3,4,5] => [5,6,7,4,1,2,3] => ? = 2
[8,6,5,7,4,1,2,3] => [1,2,8,7,5,3,6,4] => [1,2,7,5,3,6,4] => [7,5,4,6,1,2,3] => ? = 3
[5,6,7,8,4,1,2,3] => [1,2,8,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1
[6,7,8,4,5,1,2,3] => [1,2,8,3,7,4,5,6] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[8,7,5,4,6,1,2,3] => [1,2,8,6,3,7,5,4] => [1,2,6,3,7,5,4] => [7,6,4,1,2,3,5] => ? = 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: St000325
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 87%●distinct values known / distinct values provided: 60%
Mp00252: Permutations —restriction⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 87%●distinct values known / distinct values provided: 60%
Values
[1] => [1] => [] => [] => ? = 0 + 1
[1,2] => [1,2] => [1] => [1] => 1 = 0 + 1
[2,1] => [2,1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2] => [1,2] => 1 = 0 + 1
[1,3,2] => [2,3,1] => [2,1] => [2,1] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,1,2] => [1,3,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [3,2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [2,3,4,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,3,2,4] => [2,3,1,4] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,3,4,2] => [2,4,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,4,2,3] => [2,1,4,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,4,3,2] => [3,4,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,1,4,3] => [3,2,4,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,4,1,3] => [1,3,4,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[2,4,3,1] => [4,2,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,1,4,2] => [3,4,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[3,4,1,2] => [1,4,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[3,4,2,1] => [4,3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,1,2,3] => [1,2,4,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [2,4,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[4,2,1,3] => [3,1,4,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,2,3,1] => [4,1,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,3,1,2] => [1,4,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,5,3] => [2,3,5,1,4] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,5,3,4] => [2,3,1,5,4] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,3,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,2,5,4] => [3,4,2,5,1] => [3,4,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[1,3,4,5,2] => [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,3,5,2,4] => [2,1,4,5,3] => [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,3,5,4,2] => [3,5,2,4,1] => [3,2,4,1] => [2,1,4,3] => 3 = 2 + 1
[1,4,2,3,5] => [2,1,4,3,5] => [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,2,5,3] => [3,4,5,1,2] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,4,3,2,5] => [3,4,2,1,5] => [3,4,2,1] => [1,4,3,2] => 3 = 2 + 1
[1,4,3,5,2] => [3,5,2,1,4] => [3,2,1,4] => [3,2,1,4] => 3 = 2 + 1
[1,4,5,2,3] => [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,4,5,3,2] => [3,5,4,1,2] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5 + 1
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 4 + 1
[5,6,7,8,4,3,2,1] => [8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3 + 1
[4,5,6,7,8,3,2,1] => [8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2 + 1
[4,5,6,7,3,8,2,1] => [8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 2 + 1
[4,5,6,3,7,8,2,1] => [8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => ? = 2 + 1
[5,4,3,6,7,8,2,1] => [8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => ? = 3 + 1
[3,4,5,6,7,8,2,1] => [8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1 + 1
[4,5,6,7,8,2,3,1] => [8,1,7,2,3,4,5,6] => [1,7,2,3,4,5,6] => [3,4,5,6,7,1,2] => ? = 1 + 1
[6,7,8,5,2,3,4,1] => [8,1,2,7,6,3,4,5] => [1,2,7,6,3,4,5] => [5,6,7,4,1,2,3] => ? = 2 + 1
[8,5,6,7,2,3,4,1] => [8,1,2,7,3,4,6,5] => [1,2,7,3,4,6,5] => [7,4,5,6,1,2,3] => ? = 2 + 1
[5,6,7,8,2,3,4,1] => [8,1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1 + 1
[8,7,6,2,3,4,5,1] => [8,1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3 + 1
[6,7,8,2,3,4,5,1] => [8,1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1 + 1
[8,7,2,3,4,5,6,1] => [8,1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 2 + 1
[7,8,2,3,4,5,6,1] => [8,1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1 + 1
[8,2,3,4,5,6,7,1] => [8,1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[7,6,5,4,3,2,8,1] => [8,6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 5 + 1
[7,6,5,2,3,4,8,1] => [8,1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 3 + 1
[7,6,2,3,4,5,8,1] => [8,1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 2 + 1
[5,4,3,2,6,7,8,1] => [8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 3 + 1
[3,4,5,2,6,7,8,1] => [8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 1 + 1
[5,4,2,3,6,7,8,1] => [8,1,4,3,2,5,6,7] => [1,4,3,2,5,6,7] => [4,3,1,2,5,6,7] => ? = 2 + 1
[4,3,2,5,6,7,8,1] => [8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 2 + 1
[3,4,2,5,6,7,8,1] => [8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 1 + 1
[3,2,4,5,6,7,8,1] => [8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 1 + 1
[2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0 + 1
[8,7,6,5,4,3,1,2] => [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,1,2] => ? = 5 + 1
[7,8,5,6,3,4,1,2] => [1,8,2,7,3,6,4,5] => [1,2,7,3,6,4,5] => [6,7,4,5,1,2,3] => ? = 2 + 1
[7,5,6,8,3,4,1,2] => [1,8,2,7,3,5,4,6] => [1,2,7,3,5,4,6] => [6,4,5,7,1,2,3] => ? = 2 + 1
[5,6,7,8,3,4,1,2] => [1,8,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1 + 1
[7,8,5,3,4,6,1,2] => [1,8,2,6,3,7,4,5] => [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => ? = 2 + 1
[7,8,3,4,5,6,1,2] => [1,8,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1 + 1
[8,3,4,5,6,7,1,2] => [1,8,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[7,5,6,3,4,8,1,2] => [1,8,2,6,3,5,4,7] => [1,2,6,3,5,4,7] => [6,4,5,1,2,3,7] => ? = 2 + 1
[5,6,7,3,4,8,1,2] => [1,8,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => ? = 1 + 1
[7,5,3,4,6,8,1,2] => [1,8,2,5,3,6,4,7] => [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => ? = 2 + 1
[7,3,4,5,6,8,1,2] => [1,8,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1 + 1
[5,6,3,4,7,8,1,2] => [1,8,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => ? = 1 + 1
[6,4,3,5,7,8,1,2] => [1,8,4,2,5,3,6,7] => [1,4,2,5,3,6,7] => [5,3,1,2,4,6,7] => ? = 2 + 1
[5,3,4,6,7,8,1,2] => [1,8,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1 + 1
[4,3,5,6,7,8,1,2] => [1,8,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1 + 1
[3,4,5,6,7,8,1,2] => [1,8,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0 + 1
[4,5,6,7,8,2,1,3] => [7,1,8,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1 + 1
[6,7,8,5,4,1,2,3] => [1,2,8,7,6,3,4,5] => [1,2,7,6,3,4,5] => [5,6,7,4,1,2,3] => ? = 2 + 1
[8,6,5,7,4,1,2,3] => [1,2,8,7,5,3,6,4] => [1,2,7,5,3,6,4] => [7,5,4,6,1,2,3] => ? = 3 + 1
[5,6,7,8,4,1,2,3] => [1,2,8,7,3,4,5,6] => [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1 + 1
[6,7,8,4,5,1,2,3] => [1,2,8,3,7,4,5,6] => [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1 + 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: St000619
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
St000619: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 50%
Values
[1] => ? = 0 + 1
[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,2] => 2 = 1 + 1
[2,1,3] => 2 = 1 + 1
[2,3,1] => 1 = 0 + 1
[3,1,2] => 1 = 0 + 1
[3,2,1] => 2 = 1 + 1
[1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => 3 = 2 + 1
[2,1,3,4] => 2 = 1 + 1
[2,1,4,3] => 3 = 2 + 1
[2,3,1,4] => 2 = 1 + 1
[2,3,4,1] => 1 = 0 + 1
[2,4,1,3] => 2 = 1 + 1
[2,4,3,1] => 2 = 1 + 1
[3,1,2,4] => 2 = 1 + 1
[3,1,4,2] => 2 = 1 + 1
[3,2,1,4] => 3 = 2 + 1
[3,2,4,1] => 2 = 1 + 1
[3,4,1,2] => 1 = 0 + 1
[3,4,2,1] => 2 = 1 + 1
[4,1,2,3] => 1 = 0 + 1
[4,1,3,2] => 2 = 1 + 1
[4,2,1,3] => 2 = 1 + 1
[4,2,3,1] => 2 = 1 + 1
[4,3,1,2] => 2 = 1 + 1
[4,3,2,1] => 3 = 2 + 1
[1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => 3 = 2 + 1
[1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => 3 = 2 + 1
[1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => 3 = 2 + 1
[1,4,3,5,2] => 3 = 2 + 1
[1,4,5,2,3] => 2 = 1 + 1
[1,4,5,3,2] => 3 = 2 + 1
[1,4,6,5,7,2,3] => ? = 2 + 1
[1,4,6,5,7,3,2] => ? = 3 + 1
[1,4,6,7,2,3,5] => ? = 1 + 1
[1,4,6,7,2,5,3] => ? = 2 + 1
[1,4,6,7,3,2,5] => ? = 2 + 1
[1,4,6,7,3,5,2] => ? = 2 + 1
[1,4,6,7,5,2,3] => ? = 2 + 1
[1,4,6,7,5,3,2] => ? = 3 + 1
[1,4,7,2,3,5,6] => ? = 1 + 1
[1,4,7,2,3,6,5] => ? = 2 + 1
[1,4,7,2,5,3,6] => ? = 2 + 1
[1,4,7,2,5,6,3] => ? = 2 + 1
[1,4,7,2,6,3,5] => ? = 2 + 1
[1,4,7,2,6,5,3] => ? = 3 + 1
[1,4,7,3,2,5,6] => ? = 2 + 1
[1,4,7,3,2,6,5] => ? = 3 + 1
[1,4,7,3,5,2,6] => ? = 2 + 1
[1,4,7,3,5,6,2] => ? = 2 + 1
[1,4,7,3,6,2,5] => ? = 2 + 1
[1,4,7,3,6,5,2] => ? = 3 + 1
[1,4,7,5,2,3,6] => ? = 2 + 1
[1,4,7,5,2,6,3] => ? = 3 + 1
[1,4,7,5,3,2,6] => ? = 3 + 1
[1,4,7,5,3,6,2] => ? = 3 + 1
[1,4,7,5,6,2,3] => ? = 2 + 1
[1,4,7,5,6,3,2] => ? = 3 + 1
[1,4,7,6,2,3,5] => ? = 2 + 1
[1,4,7,6,2,5,3] => ? = 3 + 1
[1,4,7,6,3,2,5] => ? = 3 + 1
[1,4,7,6,3,5,2] => ? = 3 + 1
[1,4,7,6,5,2,3] => ? = 3 + 1
[1,4,7,6,5,3,2] => ? = 4 + 1
[1,5,2,3,4,6,7] => ? = 1 + 1
[1,5,2,3,4,7,6] => ? = 2 + 1
[1,5,2,3,6,4,7] => ? = 2 + 1
[1,5,2,3,6,7,4] => ? = 2 + 1
[1,5,2,3,7,4,6] => ? = 2 + 1
[1,5,2,3,7,6,4] => ? = 3 + 1
[1,5,2,4,3,6,7] => ? = 2 + 1
[1,5,2,4,3,7,6] => ? = 3 + 1
[1,5,2,4,6,3,7] => ? = 2 + 1
[1,5,2,4,6,7,3] => ? = 2 + 1
[1,5,2,4,7,3,6] => ? = 2 + 1
[1,5,2,4,7,6,3] => ? = 3 + 1
[1,5,2,6,3,4,7] => ? = 2 + 1
[1,5,2,6,3,7,4] => ? = 3 + 1
[1,5,2,6,4,3,7] => ? = 3 + 1
[1,5,2,6,4,7,3] => ? = 3 + 1
[1,5,2,6,7,3,4] => ? = 2 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
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