- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 12061 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

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 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.