- FindStat

Your data matches 60 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000662
(load all 3 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,1,3] => 1
[1,3,2] => [1,3,2] => [2,1,3] => 1
[2,1,3] => [2,1,3] => [3,2,1] => 2
[2,3,1] => [2,3,1] => [3,1,2] => 1
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => 1
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => 1
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => 1
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 3
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 2
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => 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: St000288
(load all 3 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] =>  => ? = 0
[1,2] => [1,2] => [2,1] => 1 => 1
[2,1] => [2,1] => [1,2] => 0 => 0
[1,2,3] => [1,3,2] => [2,1,3] => 10 => 1
[1,3,2] => [1,3,2] => [2,1,3] => 10 => 1
[2,1,3] => [2,1,3] => [3,2,1] => 11 => 2
[2,3,1] => [2,3,1] => [3,1,2] => 10 => 1
[3,1,2] => [3,1,2] => [1,3,2] => 01 => 1
[3,2,1] => [3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => 100 => 1
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => 100 => 1
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => 100 => 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => 100 => 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => 100 => 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => 100 => 1
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => 110 => 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 110 => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 101 => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 100 => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 101 => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 100 => 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => 110 => 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => 110 => 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 111 => 3
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 110 => 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 101 => 2
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 100 => 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => 010 => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 011 => 2
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 010 => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 001 => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,3,4,5] => 1000 => 1
[] => [] => [] => ? => ? = 0

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St001176

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1]
 => 0
[1,2] => [1,2] => [2,1] => [1,1]
 => 1
[2,1] => [2,1] => [1,2] => [2]
 => 0
[1,2,3] => [1,3,2] => [2,1,3] => [2,1]
 => 1
[1,3,2] => [1,3,2] => [2,1,3] => [2,1]
 => 1
[2,1,3] => [2,1,3] => [3,2,1] => [1,1,1]
 => 2
[2,3,1] => [2,3,1] => [3,1,2] => [2,1]
 => 1
[3,1,2] => [3,1,2] => [1,3,2] => [2,1]
 => 1
[3,2,1] => [3,2,1] => [1,2,3] => [3]
 => 0
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => [3,1]
 => 1
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => [3,1]
 => 1
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => [3,1]
 => 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => [3,1]
 => 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => [3,1]
 => 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => [3,1]
 => 1
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => [2,1,1]
 => 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [2,1,1]
 => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [2,2]
 => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => [3,1]
 => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [2,2]
 => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => [3,1]
 => 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [2,1,1]
 => 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => [2,1,1]
 => 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => [1,1,1,1]
 => 3
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => [2,1,1]
 => 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [2,1,1]
 => 2
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => [3,1]
 => 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => [3,1]
 => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [3,1]
 => 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => [2,1,1]
 => 2
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => [3,1]
 => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [3,1]
 => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [4]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => [4,1]
 => 1
[] => [] => [] => []
 => ? = 0
[8,7,6,9,5,4,3,2,1] => [8,7,6,9,5,4,3,2,1] => [9,8,7,1,2,3,4,5,6] => ?
 => ? = 3
[8,7,6,5,9,4,3,2,1] => [8,7,6,5,9,4,3,2,1] => [9,8,7,6,1,2,3,4,5] => ?
 => ? = 4
[8,7,6,5,4,9,3,2,1] => [8,7,6,5,4,9,3,2,1] => [9,8,7,6,5,1,2,3,4] => ?
 => ? = 5
[8,7,6,5,4,3,9,2,1] => [8,7,6,5,4,3,9,2,1] => [9,8,7,6,5,4,1,2,3] => ?
 => ? = 6
[9,8,7,6,10,5,4,3,2,1] => [9,8,7,6,10,5,4,3,2,1] => [10,9,8,7,1,2,3,4,5,6] => ?
 => ? = 4
[9,8,7,6,5,10,4,3,2,1] => [9,8,7,6,5,10,4,3,2,1] => [10,9,8,7,6,1,2,3,4,5] => ?
 => ? = 5
[9,8,7,6,5,4,10,3,2,1] => [9,8,7,6,5,4,10,3,2,1] => [10,9,8,7,6,5,1,2,3,4] => ?
 => ? = 6
[9,8,7,6,5,4,3,10,2,1] => [9,8,7,6,5,4,3,10,2,1] => [10,9,8,7,6,5,4,1,2,3] => ?
 => ? = 7

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000157

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [[1]]
 => 0
[1,2] => [1,2] => [2,1] => [[1],[2]]
 => 1
[2,1] => [2,1] => [1,2] => [[1,2]]
 => 0
[1,2,3] => [1,3,2] => [2,1,3] => [[1,3],[2]]
 => 1
[1,3,2] => [1,3,2] => [2,1,3] => [[1,3],[2]]
 => 1
[2,1,3] => [2,1,3] => [3,2,1] => [[1],[2],[3]]
 => 2
[2,3,1] => [2,3,1] => [3,1,2] => [[1,3],[2]]
 => 1
[3,1,2] => [3,1,2] => [1,3,2] => [[1,2],[3]]
 => 1
[3,2,1] => [3,2,1] => [1,2,3] => [[1,2,3]]
 => 0
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => [[1,3,4],[2]]
 => 1
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => [[1,4],[2],[3]]
 => 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [[1,4],[2],[3]]
 => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
 => 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
 => 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => [[1],[2],[3],[4]]
 => 3
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => [[1,4],[2],[3]]
 => 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [[1,3],[2],[4]]
 => 2
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => [[1,2],[3],[4]]
 => 2
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => [[1,2,4],[3]]
 => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => [[1,3,4,5],[2]]
 => 1
[1,2,3,4,5,6,7,8,9,11,10] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,10,9,8,7,6,5,4,3,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,2,3,4,5,6,7,8,9,10] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,9,8,7,6,5,4,3,11,10,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,8,7,6,5,4,3,2,10,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[10,11,9,8,7,6,5,4,3,2,1] => [10,11,9,8,7,6,5,4,3,2,1] => [11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[11,10,9,8,7,6,5,4,3,1,2] => [11,10,9,8,7,6,5,4,3,1,2] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
 => ? = 1
[1,3,4,5,6,7,8,9,10,11,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[11,9,8,7,6,5,4,3,2,1,10] => [11,9,8,7,6,5,4,3,2,1,10] => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
 => ? = 9
[9,8,7,6,5,4,3,2,1,10,11] => [9,8,7,6,5,4,3,2,1,11,10] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 9
[1,11,10,9,8,7,6,5,2,4,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,10,9,8,7,6,2,5,4,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,10,9,8,2,7,6,5,4,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,10,9,2,8,7,6,5,4,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,2,10,9,8,7,6,5,4,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,11,10,9,8,7,6,5,4,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,7,8,9,10,11,2,3,4,5,6] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,6,8,5,7,10,9,11] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,7,5,8,6,10,9,11] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,6,7,8,10,11,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,9,11,10,8,7,6,5,4,3,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,8,11,10,9,7,6,5,4,3,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,6,11,10,9,8,7,5,4,3,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,5,11,10,9,8,7,6,4,3,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,3,11,10,9,8,7,6,5,4,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,11,2,3,4,5,6,7,8,10,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,11,3,4,5,6,7,8,10,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,10,4,5,6,7,11,8,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,11,5,6,7,8,10,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,11,6,7,8,10,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,6,10,7,11,8,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,6,7,11,8,10,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,6,7,8,11,10,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,6,7,8,11,9,10] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,6,7,8,9,10,11] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,6,7,8,10,9,11] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[9,8,7,6,5,4,3,2,1,11,10] => [9,8,7,6,5,4,3,2,1,11,10] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? = 9
[1,3,2,5,4,7,6,9,8,11,10] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,7,6,8,5,9,4,10,3,11,2] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,4,5,6,7,11,8,9,2,10,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[9,11,10,8,7,6,5,4,3,2,1] => [9,11,10,8,7,6,5,4,3,2,1] => [10,1,2,3,4,5,6,7,8,9,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,3,4,5,6,7,8,9,11,2,10] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,4,2,5,6,7,8,9,11,10,3] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,10,3,4,5,6,7,8,11,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[1,2,3,4,5,10,6,7,8,11,9] => [1,11,10,9,8,7,6,5,4,3,2] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
 => ? = 1
[11,9,10,8,7,6,5,4,3,2,1] => [11,9,10,8,7,6,5,4,3,2,1] => [1,11,2,3,4,5,6,7,8,9,10] => [[1,2,4,5,6,7,8,9,10,11],[3]]
 => ? = 1

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: St000996
(load all 11 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [3,1,2] => 1
[1,3,2] => [1,3,2] => [3,1,2] => 1
[2,1,3] => [2,1,3] => [2,3,1] => 2
[2,3,1] => [2,3,1] => [2,1,3] => 1
[3,1,2] => [3,1,2] => [1,3,2] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => 1
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => 1
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => 1
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[4,3,2,1,5,6,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 4
[4,3,2,1,5,7,6] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 4
[4,3,2,1,6,5,7] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 4
[4,3,2,1,6,7,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 4
[4,3,2,1,7,5,6] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 4
[4,3,2,1,7,6,5] => [4,3,2,1,7,6,5] => [4,5,6,7,1,2,3] => ? = 4
[4,5,3,6,7,1,2] => [4,7,3,6,5,1,2] => [4,1,5,2,3,7,6] => ? = 3
[4,5,3,6,7,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[4,5,3,7,6,1,2] => [4,7,3,6,5,1,2] => [4,1,5,2,3,7,6] => ? = 3
[4,5,3,7,6,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[4,5,6,2,3,7,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,5,6,2,7,3,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,5,6,7,1,2,3] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,5,6,7,1,3,2] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,5,6,7,2,3,1] => [4,7,6,5,2,3,1] => [4,1,2,3,6,5,7] => ? = 2
[4,5,7,2,3,6,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,5,7,2,6,3,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,5,7,6,1,2,3] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,5,7,6,1,3,2] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,5,7,6,2,3,1] => [4,7,6,5,2,3,1] => [4,1,2,3,6,5,7] => ? = 2
[4,6,3,5,7,1,2] => [4,7,3,6,5,1,2] => [4,1,5,2,3,7,6] => ? = 3
[4,6,3,5,7,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[4,6,3,7,5,1,2] => [4,7,3,6,5,1,2] => [4,1,5,2,3,7,6] => ? = 3
[4,6,3,7,5,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[4,6,5,2,3,7,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,6,5,2,7,3,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,6,5,7,1,2,3] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,6,5,7,1,3,2] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,6,5,7,2,3,1] => [4,7,6,5,2,3,1] => [4,1,2,3,6,5,7] => ? = 2
[4,6,7,2,3,5,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,6,7,2,5,3,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,6,7,5,1,2,3] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,6,7,5,1,3,2] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,6,7,5,2,3,1] => [4,7,6,5,2,3,1] => [4,1,2,3,6,5,7] => ? = 2
[4,7,3,5,6,1,2] => [4,7,3,6,5,1,2] => [4,1,5,2,3,7,6] => ? = 3
[4,7,3,5,6,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[4,7,3,6,5,1,2] => [4,7,3,6,5,1,2] => [4,1,5,2,3,7,6] => ? = 3
[4,7,3,6,5,2,1] => [4,7,3,6,5,2,1] => [4,1,5,2,3,6,7] => ? = 2
[4,7,5,2,3,6,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,7,5,2,6,3,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,7,5,6,1,2,3] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,7,5,6,1,3,2] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,7,5,6,2,3,1] => [4,7,6,5,2,3,1] => [4,1,2,3,6,5,7] => ? = 2
[4,7,6,2,3,5,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,7,6,2,5,3,1] => [4,7,6,2,5,3,1] => [4,1,2,6,3,5,7] => ? = 2
[4,7,6,5,1,2,3] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,7,6,5,1,3,2] => [4,7,6,5,1,3,2] => [4,1,2,3,7,5,6] => ? = 2
[4,7,6,5,2,3,1] => [4,7,6,5,2,3,1] => [4,1,2,3,6,5,7] => ? = 2
[5,6,7,3,4,2,1] => [5,7,6,3,4,2,1] => [3,1,2,5,4,6,7] => ? = 2
[5,6,7,4,2,3,1] => [5,7,6,4,2,3,1] => [3,1,2,4,6,5,7] => ? = 2

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

Matching statistic: St001581

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 87%●distinct values known / distinct values provided: 70%

Values

[1] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[2,1] => [2,1] => [1,2] => ([],2)
 => 1 = 0 + 1
[1,2,3] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[1,3,2] => [1,3,2] => [2,1,3] => ([(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,3,1] => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
 => 1 = 0 + 1
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
 => 2 = 1 + 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => ([(2,3)],4)
 => 2 = 1 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([],4)
 => 1 = 0 + 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(3,4)],5)
 => 2 = 1 + 1
[7,6,5,4,8,3,2,1] => [7,6,5,4,8,3,2,1] => [8,7,6,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[7,6,5,4,3,8,2,1] => [7,6,5,4,3,8,2,1] => [8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,4,8,2,3,1] => [7,6,5,4,8,2,3,1] => [8,7,6,5,1,4,2,3] => ([(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,8,5,4,3,2,6,1] => [7,8,5,4,3,2,6,1] => [8,1,7,6,5,4,2,3] => ([(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,5,4,3,2,7,1] => [8,6,5,4,3,2,7,1] => [1,8,7,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,4,3,2,8,1] => [7,6,5,4,3,2,8,1] => [8,7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,8,5,4,3,6,1,2] => [7,8,5,4,3,6,1,2] => [8,1,7,6,5,2,4,3] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,8,4,5,2,1,3] => [7,6,8,4,5,2,1,3] => [8,7,1,6,2,5,4,3] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,8,2,1,3,4] => [7,6,5,8,2,1,4,3] => [8,7,6,1,4,3,2,5] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,5,4,3,2,1,7] => [8,6,5,4,3,2,1,7] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[6,7,5,4,3,2,1,8] => [6,8,5,4,3,2,1,7] => [7,1,8,6,5,4,3,2] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[5,6,4,7,3,2,1,8] => [5,8,4,7,3,2,1,6] => [6,1,7,2,8,5,4,3] => ([(0,7),(1,3),(1,7),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,6,4,2,3,1,8] => [7,5,8,4,2,6,1,3] => [8,6,1,7,4,2,5,3] => ([(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,4,3,1,2,8] => [7,6,5,4,3,1,8,2] => [8,7,6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,7,5,4,3,1,2,8] => [6,8,5,4,3,1,7,2] => [7,1,8,6,5,3,2,4] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,7,4,2,1,3,8] => [6,5,8,4,2,1,7,3] => [7,6,1,8,4,3,2,5] => ([(0,6),(0,7),(1,2),(1,6),(1,7),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,3,2,1,4,8] => [7,6,5,3,2,1,8,4] => [8,7,6,4,3,2,1,5] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,5,4,3,2,1,6,8] => [7,5,4,3,2,1,8,6] => [8,6,5,4,3,2,1,7] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,5,3,2,1,4,6,8] => [7,5,3,2,1,8,6,4] => [8,6,4,3,2,1,5,7] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,3,1,2,4,6,8] => [7,5,3,1,8,6,4,2] => [8,6,4,2,1,3,5,7] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4 + 1
[6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[5,4,3,2,1,6,7,8] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[5,4,3,2,1,8,7,6] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[5,4,3,2,1,7,8,6] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,4,3,2,1,8,7] => [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[5,4,3,2,1,6,8,7] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[5,4,3,2,1,7,6,8] => [5,4,3,2,1,8,7,6] => [6,5,4,3,2,1,7,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[2,1,4,3,6,5,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,3,2,1,6,5,8,7,10,9] => [4,3,2,1,10,9,8,7,6,5] => [5,4,3,2,1,6,7,8,9,10] => ([(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4 + 1
[2,1,5,6,3,4,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,6,5,4,3,8,7,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[6,5,4,3,2,1,8,7,10,9] => [6,5,4,3,2,1,10,9,8,7] => [7,6,5,4,3,2,1,8,9,10] => ([(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 6 + 1
[2,1,8,5,4,7,6,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,10,5,4,7,6,9,8,3] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,6,7,8,3,4,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,5,7,3,8,4,6,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,4,3,7,8,5,6,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,4,3,8,7,6,5,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,3,2,1,8,7,6,5,10,9] => [4,3,2,1,10,9,8,7,6,5] => [5,4,3,2,1,6,7,8,9,10] => ([(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4 + 1
[2,1,8,7,6,5,4,3,10,9] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[8,7,6,5,4,3,2,1,10,9] => [8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,1,10] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 8 + 1
[2,1,10,7,6,5,4,9,8,3] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,4,3,10,7,6,9,8,5] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[4,3,2,1,10,7,6,9,8,5] => [4,3,2,1,10,9,8,7,6,5] => [5,4,3,2,1,6,7,8,9,10] => ([(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 4 + 1
[2,1,10,9,6,5,8,7,4,3] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,7,8,9,10,3,4,5,6] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,6,8,9,3,10,4,5,7] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,5,8,3,9,10,4,6,7] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1
[2,1,4,3,8,9,10,5,6,7] => [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => ([(7,8),(7,9),(8,9)],10)
 => ? = 2 + 1

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

Matching statistic: St000374
(load all 15 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,3,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,6,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,6,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,7,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,7,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,4,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,5,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,7,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,7,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,4,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,4,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,5,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,5,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,6,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,6,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,3,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,6,3,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,6,7,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,7,3,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,7,6,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,3,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,3,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,5,7,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,7,3,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,7,5,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,3,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,3,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,5,3,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,5,6,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,6,3,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,6,5,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,5,3,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,5,3,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000703
(load all 8 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,3,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,6,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,6,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,7,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,7,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,4,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,5,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,7,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,7,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,4,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,4,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,5,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,5,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,6,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,6,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,3,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,6,3,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,6,7,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,7,3,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,5,7,6,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,3,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,3,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,5,7,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,7,3,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,6,7,5,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,3,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,3,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,5,3,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,5,6,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,6,3,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,7,6,5,1,3] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,5,3,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,5,3,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 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]].

Matching statistic: St000337
(load all 4 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,3,1] => 1
[1,3,2] => [1,3,2] => [2,3,1] => 1
[2,1,3] => [2,1,3] => [3,1,2] => 2
[2,3,1] => [2,3,1] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => 2
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => 1
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 3
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => 1
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 1
[2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,5,6,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,5,7,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,6,5,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,6,7,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,4,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,7,5,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,4,7,6,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,4,6,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,5,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,4,7,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,5,6,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,6,4,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,5,6,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,6,7,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,5,7,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,7,4,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,5,7,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,5,7,6,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,4,5,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,6,4,7,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,4,7,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,5,4,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,6,5,7,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,5,7,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,6,7,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,7,4,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,6,7,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,6,7,5,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,7,4,5,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,4,5,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,7,4,6,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,4,6,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,7,5,4,1,6] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,5,4,6,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,7,5,6,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,5,6,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,7,6,4,1,5] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,6,4,5,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,3,7,6,5,1,4] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,3,7,6,5,4,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1
[2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,4,5,6,7,2] => ? = 2
[2,4,3,5,6,7,1] => [2,7,6,5,4,3,1] => [1,3,4,5,6,7,2] => ? = 1

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$ where

$\textrm{inv} \, \tau_{i}$ is the number of inversions of

$\tau_{i}$ .

Matching statistic: St000141
(load all 2 compositions to match this statistic)

Mapping

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[1,3,2] => [1,3,2] => [2,1,3] => [2,1,3] => 1
[2,1,3] => [2,1,3] => [3,2,1] => [3,2,1] => 2
[2,3,1] => [2,3,1] => [3,1,2] => [1,3,2] => 1
[3,1,2] => [3,1,2] => [1,3,2] => [2,3,1] => 1
[3,2,1] => [3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,4,3,2] => [2,1,3,4] => [2,1,3,4] => 1
[1,2,4,3] => [1,4,3,2] => [2,1,3,4] => [2,1,3,4] => 1
[1,3,2,4] => [1,4,3,2] => [2,1,3,4] => [2,1,3,4] => 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => [2,1,3,4] => 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => [2,1,3,4] => 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,3,4] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => [1,3,2,4] => 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [3,4,1,2] => 2
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => [1,3,2,4] => 1
[3,1,2,4] => [3,1,4,2] => [4,2,1,3] => [3,1,4,2] => 2
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => [3,1,4,2] => 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 3
[3,2,4,1] => [3,2,4,1] => [4,3,1,2] => [1,4,3,2] => 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => [2,4,3,1] => 2
[3,4,2,1] => [3,4,2,1] => [4,1,2,3] => [1,2,4,3] => 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => [2,3,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [1,4,3,2] => [3,4,2,1] => 2
[4,2,3,1] => [4,2,3,1] => [1,4,2,3] => [2,1,4,3] => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => [2,3,4,1] => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[2,3,4,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,4,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,4,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,4,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,4,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,4,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,5,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,5,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,5,6,4,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,5,6,7,1,4] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,5,7,4,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,5,7,6,1,4] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,6,4,5,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,6,4,7,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,6,5,4,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,6,5,7,1,4] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,6,7,4,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,6,7,5,1,4] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,7,4,5,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,7,4,6,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,7,5,4,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,7,5,6,1,4] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,7,6,4,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,3,7,6,5,1,4] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,3,5,6,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,3,5,7,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,3,6,5,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,3,6,7,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,3,7,5,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,3,7,6,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,5,3,6,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,5,3,7,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,5,6,3,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,5,6,7,1,3] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,5,7,3,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,5,7,6,1,3] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,6,3,5,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,6,3,7,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,6,5,3,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,6,5,7,1,3] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,6,7,3,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,6,7,5,1,3] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,7,3,5,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,7,3,6,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,7,5,3,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,7,5,6,1,3] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,7,6,3,1,5] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,4,7,6,5,1,3] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,5,3,4,6,1,7] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2
[2,5,3,4,7,1,6] => [2,7,6,5,4,1,3] => [3,1,2,4,5,7,6] => [2,4,3,5,6,7,1] => ? = 2

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000028The number of stack-sorts needed to sort a permutation. St000245The number of ascents of a permutation. St000054The first entry of the permutation. St000451The length of the longest pattern of the form k 1 2. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000362The size of a minimal vertex cover of a graph. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001670The connected partition number of a graph. St001489The maximum of the number of descents and the number of inverse descents. St000829The Ulam distance of a permutation to the identity permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000740The last entry of a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000702The number of weak deficiencies of a permutation. St000446The disorder of a permutation. St000809The reduced reflection length of the permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000021The number of descents of a permutation. St000260The radius of a connected graph. St000542The number of left-to-right-minima of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000305The inverse major index of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001497The position of the largest weak excedence of a permutation. St001346The number of parking functions that give the same permutation. St000216The absolute length of a permutation. St000990The first ascent of a permutation. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001330The hat guessing number of a graph. St000710The number of big deficiencies of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001624The breadth of a lattice.