- FindStat

Identifier

Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000662: Permutations ⟶ ℤ

Values

[1] => [1] => 0

[1,2] => [1,2] => 0

[2,1] => [2,1] => 1

[1,2,3] => [1,2,3] => 0

[1,3,2] => [2,3,1] => 1

[2,1,3] => [2,1,3] => 1

[2,3,1] => [3,1,2] => 1

[3,1,2] => [1,3,2] => 1

[3,2,1] => [3,2,1] => 2

[1,2,3,4] => [1,2,3,4] => 0

[1,2,4,3] => [2,3,4,1] => 1

[1,3,2,4] => [2,3,1,4] => 1

[1,3,4,2] => [2,4,1,3] => 1

[1,4,2,3] => [2,1,4,3] => 1

[1,4,3,2] => [3,4,2,1] => 2

[2,1,3,4] => [2,1,3,4] => 1

[2,1,4,3] => [3,2,4,1] => 2

[2,3,1,4] => [3,1,2,4] => 1

[2,3,4,1] => [4,1,2,3] => 1

[2,4,1,3] => [1,3,4,2] => 1

[2,4,3,1] => [4,2,3,1] => 2

[3,1,2,4] => [1,3,2,4] => 1

[3,1,4,2] => [3,4,1,2] => 2

[3,2,1,4] => [3,2,1,4] => 2

[3,2,4,1] => [4,2,1,3] => 2

[3,4,1,2] => [1,4,2,3] => 1

[3,4,2,1] => [4,3,1,2] => 2

[4,1,2,3] => [1,2,4,3] => 1

[4,1,3,2] => [2,4,3,1] => 2

[4,2,1,3] => [3,1,4,2] => 2

[4,2,3,1] => [4,1,3,2] => 2

[4,3,1,2] => [1,4,3,2] => 2

[4,3,2,1] => [4,3,2,1] => 3

[1,2,3,4,5] => [1,2,3,4,5] => 0

[1,2,3,5,4] => [2,3,4,5,1] => 1

[1,2,4,3,5] => [2,3,4,1,5] => 1

[1,2,4,5,3] => [2,3,5,1,4] => 1

[1,2,5,3,4] => [2,3,1,5,4] => 1

[1,2,5,4,3] => [3,4,5,2,1] => 2

[1,3,2,4,5] => [2,3,1,4,5] => 1

[1,3,2,5,4] => [3,4,2,5,1] => 2

[1,3,4,2,5] => [2,4,1,3,5] => 1

[1,3,4,5,2] => [2,5,1,3,4] => 1

[1,3,5,2,4] => [2,1,4,5,3] => 1

[1,3,5,4,2] => [3,5,2,4,1] => 2

[1,4,2,3,5] => [2,1,4,3,5] => 1

[1,4,2,5,3] => [3,4,5,1,2] => 2

[1,4,3,2,5] => [3,4,2,1,5] => 2

[1,4,3,5,2] => [3,5,2,1,4] => 2

[1,4,5,2,3] => [2,1,5,3,4] => 1

[1,4,5,3,2] => [3,5,4,1,2] => 2

[1,5,2,3,4] => [2,1,3,5,4] => 1

[1,5,2,4,3] => [3,2,5,4,1] => 2

[1,5,3,2,4] => [3,4,1,5,2] => 2

[1,5,3,4,2] => [3,5,1,4,2] => 2

[1,5,4,2,3] => [3,1,5,4,2] => 2

[1,5,4,3,2] => [4,5,3,2,1] => 3

[2,1,3,4,5] => [2,1,3,4,5] => 1

[2,1,3,5,4] => [3,2,4,5,1] => 2

[2,1,4,3,5] => [3,2,4,1,5] => 2

[2,1,4,5,3] => [3,2,5,1,4] => 2

[2,1,5,3,4] => [3,2,1,5,4] => 2

[2,1,5,4,3] => [4,3,5,2,1] => 3

[2,3,1,4,5] => [3,1,2,4,5] => 1

[2,3,1,5,4] => [4,2,3,5,1] => 2

[2,3,4,1,5] => [4,1,2,3,5] => 1

[2,3,4,5,1] => [5,1,2,3,4] => 1

[2,3,5,1,4] => [1,3,4,5,2] => 1

[2,3,5,4,1] => [5,2,3,4,1] => 2

[2,4,1,3,5] => [1,3,4,2,5] => 1

[2,4,1,5,3] => [4,2,5,1,3] => 2

[2,4,3,1,5] => [4,2,3,1,5] => 2

[2,4,3,5,1] => [5,2,3,1,4] => 2

[2,4,5,1,3] => [1,3,5,2,4] => 1

[2,4,5,3,1] => [5,2,4,1,3] => 2

[2,5,1,3,4] => [1,3,2,5,4] => 1

[2,5,1,4,3] => [2,4,5,3,1] => 2

[2,5,3,1,4] => [4,2,1,5,3] => 2

[2,5,3,4,1] => [5,2,1,4,3] => 2

[2,5,4,1,3] => [1,4,5,3,2] => 2

[2,5,4,3,1] => [5,3,4,2,1] => 3

[3,1,2,4,5] => [1,3,2,4,5] => 1

[3,1,2,5,4] => [2,4,3,5,1] => 2

[3,1,4,2,5] => [3,4,1,2,5] => 2

[3,1,4,5,2] => [3,5,1,2,4] => 2

[3,1,5,2,4] => [3,1,4,5,2] => 2

[3,1,5,4,2] => [4,5,2,3,1] => 3

[3,2,1,4,5] => [3,2,1,4,5] => 2

[3,2,1,5,4] => [4,3,2,5,1] => 3

[3,2,4,1,5] => [4,2,1,3,5] => 2

[3,2,4,5,1] => [5,2,1,3,4] => 2

[3,2,5,1,4] => [1,4,3,5,2] => 2

[3,2,5,4,1] => [5,3,2,4,1] => 3

[3,4,1,2,5] => [1,4,2,3,5] => 1

[3,4,1,5,2] => [4,5,1,2,3] => 2

[3,4,2,1,5] => [4,3,1,2,5] => 2

[3,4,2,5,1] => [5,3,1,2,4] => 2

[3,4,5,1,2] => [1,5,2,3,4] => 1

[3,4,5,2,1] => [5,4,1,2,3] => 2

[3,5,1,2,4] => [1,2,4,5,3] => 1

[3,5,1,4,2] => [2,5,3,4,1] => 2

>>> Load all 2405 entries. <<<

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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 Mp00062Lehmer-code to major-code bijection to the number of descents, showing that together with the number of inversions St000018The number of inversions of a permutation. it is Euler-Mahonian.

Map

major-index to inversion-number bijection

Description

Return the permutation whose Lehmer code equals the major code of the preimage.
This map sends the major index to the number of inversions.