Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000662
Mp00252: Permutations restrictionPermutations
St000662: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [] => 0
[1,2] => [1] => 0
[2,1] => [1] => 0
[1,2,3] => [1,2] => 0
[1,3,2] => [1,2] => 0
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 0
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 2
[1,4,3,5,2] => [1,4,3,2] => 2
[1,4,5,2,3] => [1,4,2,3] => 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.
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00130: Permutations descent topsBinary words
St000288: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => => ? = 0
[1,2] => [1] => [1] => => ? = 0
[2,1] => [1] => [1] => => ? = 0
[1,2,3] => [1,2] => [1,2] => 0 => 0
[1,3,2] => [1,2] => [1,2] => 0 => 0
[2,1,3] => [2,1] => [2,1] => 1 => 1
[2,3,1] => [2,1] => [2,1] => 1 => 1
[3,1,2] => [1,2] => [1,2] => 0 => 0
[3,2,1] => [2,1] => [2,1] => 1 => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 00 => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2,4] => [1,3,2] => [3,1,2] => 01 => 1
[1,3,4,2] => [1,3,2] => [3,1,2] => 01 => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,4,3,2] => [1,3,2] => [3,1,2] => 01 => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 10 => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 10 => 1
[2,3,1,4] => [2,3,1] => [1,3,2] => 01 => 1
[2,3,4,1] => [2,3,1] => [1,3,2] => 01 => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 10 => 1
[2,4,3,1] => [2,3,1] => [1,3,2] => 01 => 1
[3,1,2,4] => [3,1,2] => [2,3,1] => 01 => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => 01 => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 11 => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => 11 => 2
[3,4,1,2] => [3,1,2] => [2,3,1] => 01 => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 11 => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[4,1,3,2] => [1,3,2] => [3,1,2] => 01 => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 10 => 1
[4,2,3,1] => [2,3,1] => [1,3,2] => 01 => 1
[4,3,1,2] => [3,1,2] => [2,3,1] => 01 => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 11 => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => 001 => 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 001 => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => 001 => 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => 010 => 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => 010 => 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => 001 => 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 001 => 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => 010 => 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 001 => 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => 001 => 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => 001 => 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => 011 => 2
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => 011 => 2
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => 001 => 1
[1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => 011 => 2
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,5,2,4,3] => [1,2,4,3] => [4,1,2,3] => 001 => 1
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [2,7,4,5,1,8,3,9,6] => ? => ? = 4
[1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [2,4,7,1,3,5,6,8] => ? => ? = 1
[1,4,2,5,3,7,6,9,8] => [1,4,2,5,3,7,6,8] => [4,2,7,1,3,5,6,8] => ? => ? = 2
[6,2,7,8,9,1,3,4,5,10] => [6,2,7,8,9,1,3,4,5] => [6,7,3,1,8,2,4,9,5] => ? => ? = 4
[6,3,2,7,8,1,4,5,9,10] => [6,3,2,7,8,1,4,5,9] => [2,6,7,3,1,4,8,5,9] => ? => ? = 3
[7,4,3,2,8,9,1,5,6,10] => [7,4,3,2,8,9,1,5,6] => [3,2,7,8,4,1,5,9,6] => ? => ? = 4
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000157
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 99% values known / values provided: 99%distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => []
=> 0
[1,2] => [1] => [1] => [[1]]
=> 0
[2,1] => [1] => [1] => [[1]]
=> 0
[1,2,3] => [1,2] => [1,2] => [[1,2]]
=> 0
[1,3,2] => [1,2] => [1,2] => [[1,2]]
=> 0
[2,1,3] => [2,1] => [2,1] => [[1],[2]]
=> 1
[2,3,1] => [2,1] => [2,1] => [[1],[2]]
=> 1
[3,1,2] => [1,2] => [1,2] => [[1,2]]
=> 0
[3,2,1] => [2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,3,2,4] => [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,4,3,2] => [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,3,1,4] => [2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[2,3,4,1] => [2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[2,4,3,1] => [2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [[1,2],[3]]
=> 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [[1,2],[3]]
=> 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[3,2,4,1] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[3,4,1,2] => [3,1,2] => [2,3,1] => [[1,2],[3]]
=> 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[4,1,3,2] => [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[4,2,3,1] => [2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [[1,2],[3]]
=> 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [5,6,1,7,2,8,3,9,4] => ?
=> ? = 4
[2,3,4,5,6,7,8,10,1,9] => [2,3,4,5,6,7,8,1,9] => [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[10,9,8,7,6,5,4,3,2,1,12,11] => [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 9
[10,8,9,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [6,7,3,8,4,1,9,5,2] => ?
=> ? = 5
[10,6,7,3,4,8,1,2,5,9] => [6,7,3,4,8,1,2,5,9] => [3,6,4,1,7,2,8,5,9] => [[1,2,5,7,9],[3,6,8],[4]]
=> ? = 4
[9,8,10,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [6,7,3,8,4,9,5,2,1] => ?
=> ? = 5
[7,4,8,2,5,9,1,3,6,10] => [7,4,8,2,5,9,1,3,6] => [2,7,4,5,1,8,3,9,6] => ?
=> ? = 4
[8,1,2,3,4,5,6,9,10,7] => [8,1,2,3,4,5,6,9,7] => [9,2,3,4,5,6,7,1,8] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? = 2
[3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,1] => [2,1,3,4,5,6,7,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 2
[8,9,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [6,7,4,2,8,5,3,1] => ?
=> ? = 5
[9,8,6,7,4,5,1,2,3] => [8,6,7,4,5,1,2,3] => [6,7,4,2,8,5,3,1] => ?
=> ? = 5
[10,6,7,8,9,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [5,6,1,7,2,8,3,9,4] => ?
=> ? = 4
[9,10,6,7,8,1,2,3,4,5] => [9,6,7,8,1,2,3,4,5] => [5,6,7,2,8,3,9,4,1] => ?
=> ? = 4
[10,9,6,7,8,1,2,3,4,5] => [9,6,7,8,1,2,3,4,5] => [5,6,7,2,8,3,9,4,1] => ?
=> ? = 4
[10,8,9,6,7,1,2,3,4,5] => [8,9,6,7,1,2,3,4,5] => [5,6,7,8,3,1,9,4,2] => ?
=> ? = 4
[10,9,8,6,7,1,2,3,4,5] => [9,8,6,7,1,2,3,4,5] => [5,6,7,8,3,9,4,2,1] => ?
=> ? = 4
[8,9,10,5,6,7,1,2,3,4] => [8,9,5,6,7,1,2,3,4] => [6,7,3,8,4,1,9,5,2] => ?
=> ? = 5
[10,9,8,5,6,7,1,2,3,4] => [9,8,5,6,7,1,2,3,4] => [6,7,3,8,4,9,5,2,1] => ?
=> ? = 5
[9,10,7,8,5,6,1,2,3,4] => [9,7,8,5,6,1,2,3,4] => [6,7,8,4,2,9,5,3,1] => ?
=> ? = 5
[10,9,7,8,5,6,1,2,3,4] => [9,7,8,5,6,1,2,3,4] => [6,7,8,4,2,9,5,3,1] => ?
=> ? = 5
[10,9,8,7,5,6,1,2,3,4] => [9,8,7,5,6,1,2,3,4] => [6,7,8,4,9,5,3,2,1] => ?
=> ? = 5
[9,10,7,8,4,5,6,1,2,3] => [9,7,8,4,5,6,1,2,3] => [7,4,8,5,2,9,6,3,1] => ?
=> ? = 6
[10,9,7,8,4,5,6,1,2,3] => [9,7,8,4,5,6,1,2,3] => [7,4,8,5,2,9,6,3,1] => ?
=> ? = 6
[10,9,8,7,4,5,6,1,2,3] => [9,8,7,4,5,6,1,2,3] => [7,4,8,5,9,6,3,2,1] => ?
=> ? = 6
[10,8,9,6,7,4,5,1,2,3] => [8,9,6,7,4,5,1,2,3] => [7,8,5,3,1,9,6,4,2] => ?
=> ? = 6
[10,9,8,6,7,4,5,1,2,3] => [9,8,6,7,4,5,1,2,3] => [7,8,5,3,9,6,4,2,1] => ?
=> ? = 6
[10,9,8,7,6,4,5,1,2,3] => [9,8,7,6,4,5,1,2,3] => [7,8,5,9,6,4,3,2,1] => ?
=> ? = 6
[1,3,2,5,4,7,6,9,8] => [1,3,2,5,4,7,6,8] => [2,4,7,1,3,5,6,8] => ?
=> ? = 1
[1,4,2,5,3,7,6,9,8] => [1,4,2,5,3,7,6,8] => [4,2,7,1,3,5,6,8] => ?
=> ? = 2
[4,2,5,1,3,6,7,8,9,10] => [4,2,5,1,3,6,7,8,9] => [2,4,1,5,3,6,7,8,9] => [[1,2,4,6,7,8,9],[3,5]]
=> ? = 2
[5,2,6,7,1,3,4,8,9,10] => [5,2,6,7,1,3,4,8,9] => [5,6,2,1,3,7,4,8,9] => [[1,2,6,8,9],[3,5,7],[4]]
=> ? = 3
[5,3,2,6,1,4,7,8,9,10] => [5,3,2,6,1,4,7,8,9] => [3,2,5,1,6,4,7,8,9] => [[1,3,5,7,8,9],[2,6],[4]]
=> ? = 3
[6,2,7,8,9,1,3,4,5,10] => [6,2,7,8,9,1,3,4,5] => [6,7,3,1,8,2,4,9,5] => ?
=> ? = 4
[6,7,3,4,8,1,2,5,9,10] => [6,7,3,4,8,1,2,5,9] => [3,6,4,1,7,2,8,5,9] => [[1,2,5,7,9],[3,6,8],[4]]
=> ? = 4
[6,3,2,7,8,1,4,5,9,10] => [6,3,2,7,8,1,4,5,9] => [2,6,7,3,1,4,8,5,9] => ?
=> ? = 3
[6,4,7,2,5,1,3,8,9,10] => [6,4,7,2,5,1,3,8,9] => [4,2,6,1,7,5,3,8,9] => [[1,3,5,8,9],[2,6],[4,7]]
=> ? = 4
[6,4,3,2,7,1,5,8,9,10] => [6,4,3,2,7,1,5,8,9] => [4,3,2,6,1,7,5,8,9] => [[1,4,6,8,9],[2,7],[3],[5]]
=> ? = 4
[7,4,3,2,8,9,1,5,6,10] => [7,4,3,2,8,9,1,5,6] => [3,2,7,8,4,1,5,9,6] => ?
=> ? = 4
[7,5,3,8,2,6,1,4,9,10] => [7,5,3,8,2,6,1,4,9] => [3,5,2,7,1,8,6,4,9] => [[1,2,4,6,9],[3,7],[5,8]]
=> ? = 4
[7,5,4,3,2,8,1,6,9,10] => [7,5,4,3,2,8,1,6,9] => [5,4,3,2,7,1,8,6,9] => [[1,5,7,9],[2,8],[3],[4],[6]]
=> ? = 5
[8,6,9,4,7,2,5,1,3,10] => [8,6,9,4,7,2,5,1,3] => [6,4,2,8,1,9,7,5,3] => [[1,4,6],[2,7],[3,8],[5,9]]
=> ? = 6
[8,6,4,3,9,2,7,1,5,10] => [8,6,4,3,9,2,7,1,5] => [4,3,6,2,8,1,9,7,5] => [[1,3,5,7],[2,8],[4,9],[6]]
=> ? = 5
[8,6,5,4,3,2,9,1,7,10] => [8,6,5,4,3,2,9,1,7] => [6,5,4,3,2,8,1,9,7] => [[1,6,8],[2,9],[3],[4],[5],[7]]
=> ? = 6
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: St000703
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00086: Permutations first fundamental transformationPermutations
St000703: Permutations ⟶ ℤResult quality: 89% values known / values provided: 89%distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => [] => 0
[1,2] => [1] => [1] => [1] => 0
[2,1] => [1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[2,3,4,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [3,1,2] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => [3,1,2] => 2
[3,4,1,2] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [3,1,2] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [3,1,2] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => [3,2,4,1] => 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => [3,2,4,1] => 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => [3,2,4,1] => 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => [2,4,3,1] => 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => [2,4,3,1] => 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => [2,4,1,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => [2,4,1,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => [2,4,3,1] => 1
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => ? = 2
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 2
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => ? = 5
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => ? = 5
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [7,1,2,6,3,4,5] => ? = 5
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [7,1,2,6,3,4,5] => ? = 5
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [6,4,2,1,7,5,3] => [6,1,7,2,3,4,5] => ? = 5
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => ? = 5
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => [7,1,2,3,5,6,4] => ? = 4
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [5,6,3,7,4,2,1] => [7,1,6,2,5,3,4] => ? = 4
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [6,7,1,2,5,3,4] => ? = 4
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [6,7,1,2,5,3,4] => ? = 4
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => ? = 3
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => ? = 3
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => ? = 3
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [7,6,1,4,5,2,3] => ? = 3
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [7,6,1,4,5,2,3] => ? = 3
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [5,6,7,4,1,2,3] => ? = 3
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => ? = 3
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [7,6,1,4,5,2,3] => ? = 3
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [5,6,7,4,1,2,3] => ? = 3
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [5,3,6,4,1,7,2] => [6,7,5,1,3,4,2] => ? = 4
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => ? = 2
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => ? = 2
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [6,7,3,4,5,1,2] => ? = 2
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,1,3,4,5,6,2] => ? = 2
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [6,7,3,4,5,1,2] => ? = 2
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => ? = 2
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => ? = 2
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [3,2,5,4,6,7,1] => [7,3,2,5,4,6,1] => ? = 3
[7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => [2,3,5,4,6,7,1] => [7,2,3,5,4,6,1] => ? = 2
[7,8,3,4,1,2,5,6] => [7,3,4,1,2,5,6] => [4,2,5,3,6,7,1] => [7,4,5,2,3,6,1] => ? = 3
[7,8,4,2,1,3,5,6] => [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => [7,5,4,3,2,6,1] => ? = 3
[8,7,4,1,2,3,5,6] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => [7,5,3,4,2,6,1] => ? = 2
[8,7,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => [7,4,2,3,5,6,1] => ? = 3
[7,8,3,2,1,4,5,6] => [7,3,2,1,4,5,6] => [4,3,2,5,6,7,1] => [7,4,2,3,5,6,1] => ? = 3
[7,8,2,1,3,4,5,6] => [7,2,1,3,4,5,6] => [3,2,4,5,6,7,1] => [7,3,2,4,5,6,1] => ? = 2
[8,7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 1
[7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => [7,2,3,4,5,6,1] => ? = 1
[8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => [4,2,3,1,5,6,7] => ? = 1
[3,2,4,5,6,7,1,8] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[7,6,5,4,3,1,2,8] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [7,1,2,3,4,6,5] => ? = 5
[7,6,5,4,1,2,3,8] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => [7,1,2,3,5,6,4] => ? = 4
[5,4,6,7,1,2,3,8] => [5,4,6,7,1,2,3] => [5,2,1,6,3,7,4] => [5,1,6,7,2,3,4] => ? = 4
[7,6,5,1,2,3,4,8] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,1,2,4,5,6,3] => ? = 3
[7,5,6,1,2,3,4,8] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [7,6,1,4,5,2,3] => ? = 3
[5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [5,6,7,4,1,2,3] => ? = 3
[6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [6,7,3,4,5,1,2] => ? = 2
[7,5,1,2,3,4,6,8] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => ? = 2
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]].
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00069: Permutations complementPermutations
St000245: Permutations ⟶ ℤResult quality: 89% values known / values provided: 89%distinct values known / distinct values provided: 100%
Values
[1] => [] => [] => [] => 0
[1,2] => [1] => [1] => [1] => 0
[2,1] => [1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => [2,1] => 0
[1,3,2] => [1,2] => [1,2] => [2,1] => 0
[2,1,3] => [2,1] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 0
[3,2,1] => [2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,3,2,4] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,4,3,2] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,3,1,4] => [2,3,1] => [1,3,2] => [3,1,2] => 1
[2,3,4,1] => [2,3,1] => [1,3,2] => [3,1,2] => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[2,4,3,1] => [2,3,1] => [1,3,2] => [3,1,2] => 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [2,1,3] => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [2,1,3] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [1,2,3] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => [1,2,3] => 2
[3,4,1,2] => [3,1,2] => [2,3,1] => [2,1,3] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 0
[4,1,3,2] => [1,3,2] => [3,1,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 1
[4,2,3,1] => [2,3,1] => [1,3,2] => [3,1,2] => 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [2,1,3] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [1,2,3] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => [1,4,3,2] => 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => [1,4,3,2] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => [1,4,3,2] => 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => [2,1,4,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => [2,1,4,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => [1,2,4,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => [1,2,4,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => [2,1,4,3] => 1
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => ? = 2
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [2,3,4,1,5,7,6] => [6,5,4,7,3,1,2] => ? = 2
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => [5,6,7,4,3,1,2] => ? = 3
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,3,2,4,5,7,6] => [7,5,6,4,3,1,2] => ? = 2
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [2,3,1,4,5,7,6] => [6,5,7,4,3,1,2] => ? = 2
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => [6,7,5,4,3,1,2] => ? = 2
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => ? = 5
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => ? = 5
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [2,4,6,1,3,5,7] => ? = 5
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [6,4,2,1,7,5,3] => [2,4,6,7,1,3,5] => ? = 5
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => ? = 2
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [5,6,3,7,4,2,1] => [3,2,5,1,4,6,7] => ? = 4
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [3,2,5,7,1,4,6] => ? = 4
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [3,6,2,5,1,4,7] => ? = 4
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [3,6,2,5,1,4,7] => ? = 4
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [3,2,5,7,1,4,6] => ? = 4
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [4,3,2,6,1,5,7] => ? = 3
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [4,3,2,6,1,5,7] => ? = 3
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [4,3,7,2,6,1,5] => ? = 3
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [4,3,2,6,1,5,7] => ? = 3
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [4,3,7,2,6,1,5] => ? = 3
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [5,3,6,4,1,7,2] => [3,5,2,4,7,1,6] => ? = 4
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [5,4,3,2,7,1,6] => ? = 2
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [5,4,3,2,7,1,6] => ? = 2
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [3,2,5,4,6,7,1] => [5,6,3,4,2,1,7] => ? = 3
[7,8,3,4,1,2,5,6] => [7,3,4,1,2,5,6] => [4,2,5,3,6,7,1] => [4,6,3,5,2,1,7] => ? = 3
[8,4,5,6,3,2,1,7] => [4,5,6,3,2,1,7] => [1,2,6,5,4,3,7] => [7,6,2,3,4,5,1] => ? = 3
[8,5,6,3,4,1,2,7] => [5,6,3,4,1,2,7] => [5,3,1,6,4,2,7] => [3,5,7,2,4,6,1] => ? = 4
[8,5,6,1,2,3,4,7] => [5,6,1,2,3,4,7] => [3,4,5,1,6,2,7] => [5,4,3,7,2,6,1] => ? = 2
[8,4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,5,6,7,3,2,1] => ? = 3
[8,2,3,4,1,5,6,7] => [2,3,4,1,5,6,7] => [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => ? = 1
[8,4,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => ? = 1
[8,3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => ? = 2
[8,2,3,1,4,5,6,7] => [2,3,1,4,5,6,7] => [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => ? = 1
[8,3,1,2,4,5,6,7] => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => [6,5,7,4,3,2,1] => ? = 1
[8,2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => ? = 1
[3,2,4,5,6,7,1,8] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => [6,7,5,4,3,1,2] => ? = 2
[7,4,5,6,1,2,3,8] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [3,6,2,5,1,4,7] => ? = 4
[5,4,6,7,1,2,3,8] => [5,4,6,7,1,2,3] => [5,2,1,6,3,7,4] => [3,6,7,2,5,1,4] => ? = 4
[7,5,6,1,2,3,4,8] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [4,3,2,6,1,5,7] => ? = 3
[5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [4,3,7,2,6,1,5] => ? = 3
[6,7,1,2,3,4,5,8] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [5,4,3,2,7,1,6] => ? = 2
[5,4,3,2,6,1,7,8] => [5,4,3,2,6,1,7] => [4,3,2,1,6,5,7] => [4,5,6,7,2,3,1] => ? = 4
[2,3,4,5,6,1,7,8] => [2,3,4,5,6,1,7] => [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => ? = 1
[5,6,3,4,1,2,7,8] => [5,6,3,4,1,2,7] => [5,3,1,6,4,2,7] => [3,5,7,2,4,6,1] => ? = 4
[4,5,6,1,2,3,7,8] => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => [4,7,3,6,2,5,1] => ? = 3
[6,1,2,3,4,5,7,8] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => ? = 1
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => [3,4,5,6,7,2,1] => ? = 4
[2,3,4,5,1,6,7,8] => [2,3,4,5,1,6,7] => [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => ? = 1
Description
The number of ascents of a permutation.
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00126: Permutations cactus evacuationPermutations
St000470: Permutations ⟶ ℤResult quality: 60% values known / values provided: 87%distinct values known / distinct values provided: 60%
Values
[1] => [] => [] => [] => ? = 0 + 1
[1,2] => [1] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [2,1,3] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [2,1,3] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [2,3,1] => [2,1,3] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [2,1,3] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 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] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,4,3,5,7,6] => [4,1,2,7,3,5,6] => ? = 2 + 1
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [2,3,4,1,5,7,6] => [2,1,3,7,4,5,6] => ? = 2 + 1
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => [3,2,4,5,7,6,1] => ? = 3 + 1
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,3,2,4,5,7,6] => [3,1,2,4,7,5,6] => ? = 2 + 1
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [2,3,1,4,5,7,6] => [2,1,3,4,7,5,6] => ? = 2 + 1
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2 + 1
[2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1 + 1
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [6,5,4,3,2,1,7] => ? = 5 + 1
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [6,5,4,3,2,1,7] => ? = 5 + 1
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [6,4,3,2,1,7,5] => ? = 5 + 1
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [6,4,2,1,7,5,3] => ? = 5 + 1
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [6,4,2,1,7,5,3] => ? = 5 + 1
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [6,4,3,2,1,7,5] => ? = 5 + 1
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [6,4,2,1,7,5,3] => ? = 5 + 1
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [6,4,2,1,7,5,3] => [6,4,2,7,5,3,1] => ? = 5 + 1
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 2 + 1
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [6,5,4,3,2,1,7] => ? = 5 + 1
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => [6,1,7,2,3,4,5] => ? = 2 + 1
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => [5,4,3,2,1,6,7] => ? = 4 + 1
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [5,6,3,7,4,2,1] => [5,3,2,1,6,4,7] => ? = 4 + 1
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [5,3,1,6,4,2,7] => ? = 4 + 1
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [5,2,1,6,3,7,4] => ? = 4 + 1
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [5,2,1,6,3,7,4] => ? = 4 + 1
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [5,3,1,6,4,2,7] => ? = 4 + 1
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 3 + 1
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 3 + 1
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 3 + 1
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [4,2,1,5,3,6,7] => ? = 3 + 1
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [4,2,1,5,3,6,7] => ? = 3 + 1
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [4,1,5,2,6,3,7] => ? = 3 + 1
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => ? = 3 + 1
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [4,2,1,5,3,6,7] => ? = 3 + 1
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [4,1,5,2,6,3,7] => ? = 3 + 1
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [5,3,6,4,1,7,2] => [5,3,6,4,1,7,2] => ? = 4 + 1
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [3,2,1,4,5,6,7] => ? = 2 + 1
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [3,2,1,4,5,6,7] => ? = 2 + 1
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [3,1,4,2,5,6,7] => ? = 2 + 1
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [3,2,1,4,5,6,7] => ? = 2 + 1
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [3,1,4,2,5,6,7] => ? = 2 + 1
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [3,2,4,1,5,6,7] => ? = 2 + 1
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [3,2,4,1,5,6,7] => ? = 2 + 1
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [3,2,5,4,6,7,1] => [3,2,5,6,1,7,4] => ? = 3 + 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: St000354
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00066: Permutations inversePermutations
St000354: Permutations ⟶ ℤResult quality: 60% values known / values provided: 86%distinct values known / distinct values provided: 60%
Values
[1] => [] => [] => [] => ? = 0
[1,2] => [1] => [1] => [1] => ? = 0
[2,1] => [1] => [1] => [1] => ? = 0
[1,2,3] => [1,2] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[2,3,4,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [3,1,2] => [2,3,1] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 2
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 2
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 1
[1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 2
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,5,2,4,3] => [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[8,7,6,5,4,3,2,1] => [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] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [2,3,4,1,5,7,6] => [4,1,2,3,5,7,6] => ? = 2
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 3
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [2,3,1,4,5,7,6] => [3,1,2,4,5,7,6] => ? = 2
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 5
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 5
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [7,6,5,2,4,1,3] => ? = 5
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [7,3,6,2,5,1,4] => ? = 5
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [7,3,6,2,5,1,4] => ? = 5
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [7,6,5,2,4,1,3] => ? = 5
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [7,3,6,2,5,1,4] => ? = 5
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [6,4,2,1,7,5,3] => [4,3,7,2,6,1,5] => ? = 5
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [7,6,5,4,3,1,2] => ? = 5
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => [7,6,5,4,1,2,3] => ? = 4
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [5,6,3,7,4,2,1] => [7,6,3,5,1,2,4] => ? = 4
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [4,7,3,6,1,2,5] => ? = 4
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [7,2,4,6,1,3,5] => ? = 4
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [7,2,4,6,1,3,5] => ? = 4
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [4,7,3,6,1,2,5] => ? = 4
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 3
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 3
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 3
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [7,4,6,1,2,3,5] => ? = 3
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [7,4,6,1,2,3,5] => ? = 3
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [3,5,7,1,2,4,6] => ? = 3
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [7,6,5,1,2,3,4] => ? = 3
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [7,4,6,1,2,3,5] => ? = 3
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [3,5,7,1,2,4,6] => ? = 3
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [5,3,6,4,1,7,2] => [5,7,2,4,1,3,6] => ? = 4
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 2
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 2
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [5,7,1,2,3,4,6] => ? = 2
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [7,6,1,2,3,4,5] => ? = 2
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [5,7,1,2,3,4,6] => ? = 2
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,5,1,2,3,4,6] => ? = 2
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,5,1,2,3,4,6] => ? = 2
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [3,2,5,4,6,7,1] => [7,2,1,4,3,5,6] => ? = 3
[7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => [2,3,5,4,6,7,1] => [7,1,2,4,3,5,6] => ? = 2
[7,8,3,4,1,2,5,6] => [7,3,4,1,2,5,6] => [4,2,5,3,6,7,1] => [7,2,4,1,3,5,6] => ? = 3
[7,8,4,2,1,3,5,6] => [7,4,2,1,3,5,6] => [4,3,5,2,6,7,1] => [7,4,2,1,3,5,6] => ? = 3
Description
The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$. In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St000021: Permutations ⟶ ℤResult quality: 60% values known / values provided: 85%distinct values known / distinct values provided: 60%
Values
[1] => [] => [] => 0
[1,2] => [1] => [1] => 0
[2,1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [3,1,2] => 1
[1,3,4,2] => [1,3,2] => [3,1,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [1,3,2] => 1
[2,3,4,1] => [2,3,1] => [1,3,2] => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2,4] => [3,1,2] => [2,3,1] => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => [2,3,1] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [3,1,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [1,3,2] => 1
[4,3,1,2] => [3,1,2] => [2,3,1] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => 2
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => 2
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => 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] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,4,3,5,7,6] => ? = 2
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [2,3,4,1,5,7,6] => ? = 2
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => ? = 3
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,3,2,4,5,7,6] => ? = 2
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [2,3,1,4,5,7,6] => ? = 2
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => ? = 2
[2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => ? = 1
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => ? = 5
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 5
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 5
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => ? = 5
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 5
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [6,4,2,1,7,5,3] => ? = 5
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => ? = 2
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => ? = 2
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 4
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [5,6,3,7,4,2,1] => ? = 4
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => ? = 4
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => ? = 4
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => ? = 4
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => ? = 4
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => ? = 3
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => ? = 3
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => ? = 3
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => ? = 3
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => ? = 3
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [5,3,6,4,1,7,2] => ? = 4
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => ? = 2
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => ? = 2
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 2
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 2
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [3,2,5,4,6,7,1] => ? = 3
[7,8,2,3,4,1,5,6] => [7,2,3,4,1,5,6] => [2,3,5,4,6,7,1] => ? = 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].
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
St000325: Permutations ⟶ ℤResult quality: 60% values known / values provided: 85%distinct values known / distinct values provided: 60%
Values
[1] => [] => [] => ? = 0 + 1
[1,2] => [1] => [1] => 1 = 0 + 1
[2,1] => [1] => [1] => 1 = 0 + 1
[1,2,3] => [1,2] => [1,2] => 1 = 0 + 1
[1,3,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1,3] => [2,1] => [2,1] => 2 = 1 + 1
[2,3,1] => [2,1] => [2,1] => 2 = 1 + 1
[3,1,2] => [1,2] => [1,2] => 1 = 0 + 1
[3,2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2,4] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[1,3,4,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,4,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[2,1,3,4] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,1,4,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1,4] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[2,3,4,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[2,4,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,4,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[3,1,2,4] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,1,4,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,2,1,4] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,2,4,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[3,4,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,4,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[4,1,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[4,2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[4,2,3,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[4,3,1,2] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[4,3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => 3 = 2 + 1
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => 2 = 1 + 1
[1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => 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] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 6 + 1
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,4,3,5,7,6] => ? = 2 + 1
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [2,3,4,1,5,7,6] => ? = 2 + 1
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => ? = 3 + 1
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,3,2,4,5,7,6] => ? = 2 + 1
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [2,3,1,4,5,7,6] => ? = 2 + 1
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => ? = 2 + 1
[2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => ? = 1 + 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
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5 + 1
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => ? = 5 + 1
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 5 + 1
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 5 + 1
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => ? = 5 + 1
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 5 + 1
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [6,4,2,1,7,5,3] => ? = 5 + 1
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => ? = 2 + 1
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 5 + 1
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => ? = 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
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [5,6,3,7,4,2,1] => ? = 4 + 1
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => ? = 4 + 1
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => ? = 4 + 1
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => ? = 4 + 1
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => ? = 4 + 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
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3 + 1
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3 + 1
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => ? = 3 + 1
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => ? = 3 + 1
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => ? = 3 + 1
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3 + 1
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => ? = 3 + 1
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => ? = 3 + 1
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [5,3,6,4,1,7,2] => ? = 4 + 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
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2 + 1
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => ? = 2 + 1
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2 + 1
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => ? = 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
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 2 + 1
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [3,2,5,4,6,7,1] => ? = 3 + 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.
Mp00252: Permutations restrictionPermutations
Mp00062: Permutations Lehmer-code to major-code bijectionPermutations
Mp00238: Permutations Clarke-Steingrimsson-ZengPermutations
St000155: Permutations ⟶ ℤResult quality: 60% values known / values provided: 85%distinct values known / distinct values provided: 60%
Values
[1] => [] => [] => [] => ? = 0
[1,2] => [1] => [1] => [1] => 0
[2,1] => [1] => [1] => [1] => 0
[1,2,3] => [1,2] => [1,2] => [1,2] => 0
[1,3,2] => [1,2] => [1,2] => [1,2] => 0
[2,1,3] => [2,1] => [2,1] => [2,1] => 1
[2,3,1] => [2,1] => [2,1] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => [1,2] => 0
[3,2,1] => [2,1] => [2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,2,4,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[1,3,4,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[1,4,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,4,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[2,1,3,4] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[2,3,4,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[2,4,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[3,1,2,4] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[3,1,4,2] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[3,2,1,4] => [3,2,1] => [3,2,1] => [2,3,1] => 2
[3,2,4,1] => [3,2,1] => [3,2,1] => [2,3,1] => 2
[3,4,1,2] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[3,4,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 2
[4,1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[4,1,3,2] => [1,3,2] => [3,1,2] => [3,1,2] => 1
[4,2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => [1,3,2] => [1,3,2] => 1
[4,3,1,2] => [3,1,2] => [2,3,1] => [3,2,1] => 1
[4,3,2,1] => [3,2,1] => [3,2,1] => [2,3,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,2,4,5,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,5,4,3] => [1,2,4,3] => [4,1,2,3] => [4,1,2,3] => 1
[1,3,2,4,5] => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[1,3,5,2,4] => [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[1,4,2,3,5] => [1,4,2,3] => [3,4,1,2] => [4,1,3,2] => 1
[1,4,2,5,3] => [1,4,2,3] => [3,4,1,2] => [4,1,3,2] => 1
[1,4,3,2,5] => [1,4,3,2] => [4,3,1,2] => [3,1,4,2] => 2
[1,4,3,5,2] => [1,4,3,2] => [4,3,1,2] => [3,1,4,2] => 2
[1,4,5,2,3] => [1,4,2,3] => [3,4,1,2] => [4,1,3,2] => 1
[1,4,5,3,2] => [1,4,3,2] => [4,3,1,2] => [3,1,4,2] => 2
[8,7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 6
[7,8,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 6
[7,6,8,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 6
[7,6,5,8,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 6
[7,6,5,4,8,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 6
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 6
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => ? = 6
[3,4,5,2,6,7,8,1] => [3,4,5,2,6,7,1] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2
[5,2,3,4,6,7,8,1] => [5,2,3,4,6,7,1] => [2,3,4,1,5,7,6] => [4,2,3,1,5,7,6] => ? = 2
[4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => [2,3,1,4,5,7,6] => ? = 3
[3,4,2,5,6,7,8,1] => [3,4,2,5,6,7,1] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 2
[4,2,3,5,6,7,8,1] => [4,2,3,5,6,7,1] => [2,3,1,4,5,7,6] => [3,2,1,4,5,7,6] => ? = 2
[3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => [2,1,3,4,5,7,6] => ? = 2
[2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 1
[8,7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => ? = 5
[7,8,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => ? = 5
[8,7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [2,3,5,7,6,4,1] => ? = 5
[8,7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [3,5,4,7,6,2,1] => ? = 5
[7,8,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [3,5,4,7,6,2,1] => ? = 5
[7,6,5,8,3,4,1,2] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => [2,3,5,7,6,4,1] => ? = 5
[7,5,6,8,3,4,1,2] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => [3,5,4,7,6,2,1] => ? = 5
[6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [6,4,2,1,7,5,3] => [2,5,4,7,6,1,3] => ? = 5
[8,3,4,5,6,7,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => ? = 2
[7,6,5,4,3,8,1,2] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => [2,3,4,5,7,6,1] => ? = 5
[3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => ? = 2
[8,7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => [2,3,4,7,5,6,1] => ? = 4
[8,7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => [5,6,3,7,4,2,1] => [2,4,7,6,5,3,1] => ? = 4
[8,6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [4,3,7,6,5,1,2] => ? = 4
[8,7,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [4,7,6,5,2,3,1] => ? = 4
[7,8,4,5,6,1,2,3] => [7,4,5,6,1,2,3] => [5,2,6,3,7,4,1] => [4,7,6,5,2,3,1] => ? = 4
[6,7,4,5,8,1,2,3] => [6,7,4,5,1,2,3] => [5,6,3,1,7,4,2] => [4,3,7,6,5,1,2] => ? = 4
[8,7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => ? = 3
[7,8,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => ? = 3
[7,6,8,5,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => ? = 3
[8,7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [3,7,6,4,5,2,1] => ? = 3
[7,8,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [3,7,6,4,5,2,1] => ? = 3
[8,5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [7,6,5,4,1,2,3] => ? = 3
[7,6,5,8,1,2,3,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => [2,3,7,4,5,6,1] => ? = 3
[7,5,6,8,1,2,3,4] => [7,5,6,1,2,3,4] => [4,5,6,2,7,3,1] => [3,7,6,4,5,2,1] => ? = 3
[5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,1,6,2,7,3] => [7,6,5,4,1,2,3] => ? = 3
[6,7,8,3,4,1,2,5] => [6,7,3,4,1,2,5] => [5,3,6,4,1,7,2] => [7,4,6,5,3,1,2] => ? = 4
[8,7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [2,7,3,4,5,6,1] => ? = 2
[7,8,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [2,7,3,4,5,6,1] => ? = 2
[8,6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [7,6,3,4,5,1,2] => ? = 2
[7,6,8,1,2,3,4,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => [2,7,3,4,5,6,1] => ? = 2
[6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => [3,4,5,6,1,7,2] => [7,6,3,4,5,1,2] => ? = 2
[8,7,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => ? = 2
[7,8,5,1,2,3,4,6] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => [7,6,3,4,5,2,1] => ? = 2
[7,8,3,2,4,1,5,6] => [7,3,2,4,1,5,6] => [3,2,5,4,6,7,1] => [7,3,2,5,4,6,1] => ? = 3
Description
The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001427The number of descents of a signed permutation. St000619The number of cyclic descents of a permutation. St001624The breadth of a lattice.