- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.