- FindStat

Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000662

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000021
(load all 3 compositions to match this statistic)

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Matching statistic: St000325
(load all 3 compositions to match this statistic)

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Matching statistic: St000470
(load all 3 compositions to match this statistic)

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => 1 = 0 + 1
[1,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,2] => [1,2] => [1,2] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,1,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,1,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,2,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[2,1,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,1,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[3,1,1] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,2,2] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[2,1,2] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,2,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[2,1,3] => [2,1,3] => [2,1,3] => 2 = 1 + 1
[2,3,1] => [3,1,2] => [2,3,1] => 2 = 1 + 1
[3,1,2] => [2,3,1] => [1,3,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,2,1] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,2,1,1] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[2,1,1,1] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,1] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,3,1,1] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[3,1,1,1] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,4,1,1] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[4,1,1,1] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,1,2] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,2,2,1] => [1,4,2,3] => [3,4,1,2] => 2 = 1 + 1
[2,1,1,2] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[2,1,2,1] => [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[2,2,1,1] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,1,3,2] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,2,1,3] => [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,2,3,1] => [1,4,2,3] => [3,4,1,2] => 2 = 1 + 1
[1,3,1,2] => [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,3,2,1] => [1,4,3,2] => [4,3,1,2] => 3 = 2 + 1
[2,1,1,3] => [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[2,1,3,1] => [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[2,3,1,1] => [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[3,1,1,2] => [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 3 = 2 + 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: St000155
(load all 3 compositions to match this statistic)

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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]].

Matching statistic: St000157

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ 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
[1,1] => [1,2] => [1,2] => [[1,2]]
 => 0
[1,2] => [1,2] => [1,2] => [[1,2]]
 => 0
[2,1] => [2,1] => [2,1] => [[1],[2]]
 => 1
[1,1,1] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,1,2] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,2,1] => [1,3,2] => [3,1,2] => [[1,3],[2]]
 => 1
[2,1,1] => [2,3,1] => [1,3,2] => [[1,2],[3]]
 => 1
[1,1,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,3,1] => [1,3,2] => [3,1,2] => [[1,3],[2]]
 => 1
[3,1,1] => [2,3,1] => [1,3,2] => [[1,2],[3]]
 => 1
[1,2,2] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[2,1,2] => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 1
[2,2,1] => [3,1,2] => [2,3,1] => [[1,2],[3]]
 => 1
[1,2,3] => [1,2,3] => [1,2,3] => [[1,2,3]]
 => 0
[1,3,2] => [1,3,2] => [3,1,2] => [[1,3],[2]]
 => 1
[2,1,3] => [2,1,3] => [2,1,3] => [[1,3],[2]]
 => 1
[2,3,1] => [3,1,2] => [2,3,1] => [[1,2],[3]]
 => 1
[3,1,2] => [2,3,1] => [1,3,2] => [[1,2],[3]]
 => 1
[3,2,1] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
 => 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,2,1] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[1,2,1,1] => [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
 => 1
[2,1,1,1] => [2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,3,1] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[1,3,1,1] => [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
 => 1
[3,1,1,1] => [2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[1,4,1,1] => [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
 => 1
[4,1,1,1] => [2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,2,1,2] => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[1,2,2,1] => [1,4,2,3] => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[2,1,1,2] => [2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[2,1,2,1] => [2,4,1,3] => [1,3,4,2] => [[1,2,3],[4]]
 => 1
[2,2,1,1] => [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
 => 0
[1,1,3,2] => [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
 => 1
[1,2,1,3] => [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
 => 1
[1,2,3,1] => [1,4,2,3] => [3,4,1,2] => [[1,2],[3,4]]
 => 1
[1,3,1,2] => [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
 => 1
[1,3,2,1] => [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
 => 2
[2,1,1,3] => [2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
 => 1
[2,1,3,1] => [2,4,1,3] => [1,3,4,2] => [[1,2,3],[4]]
 => 1
[2,3,1,1] => [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
 => 2
[3,1,1,2] => [2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
 => 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => [[1,3],[2],[4]]
 => 2

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: St000245
(load all 3 compositions to match this statistic)

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of a permutation.

Matching statistic: St000703

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Matching statistic: St000288
(load all 3 compositions to match this statistic)

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ 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
[1,1] => [1,2] => [1,2] => 0 => 0
[1,2] => [1,2] => [1,2] => 0 => 0
[2,1] => [2,1] => [2,1] => 1 => 1
[1,1,1] => [1,2,3] => [1,2,3] => 00 => 0
[1,1,2] => [1,2,3] => [1,2,3] => 00 => 0
[1,2,1] => [1,3,2] => [3,1,2] => 10 => 1
[2,1,1] => [2,3,1] => [1,3,2] => 01 => 1
[1,1,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,1] => [1,3,2] => [3,1,2] => 10 => 1
[3,1,1] => [2,3,1] => [1,3,2] => 01 => 1
[1,2,2] => [1,2,3] => [1,2,3] => 00 => 0
[2,1,2] => [2,1,3] => [2,1,3] => 10 => 1
[2,2,1] => [3,1,2] => [2,3,1] => 10 => 1
[1,2,3] => [1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => [3,1,2] => 10 => 1
[2,1,3] => [2,1,3] => [2,1,3] => 10 => 1
[2,3,1] => [3,1,2] => [2,3,1] => 10 => 1
[3,1,2] => [2,3,1] => [1,3,2] => 01 => 1
[3,2,1] => [3,2,1] => [3,2,1] => 11 => 2
[1,1,1,1] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,1,1,2] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,1,2,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,2,1,1] => [1,3,4,2] => [2,4,1,3] => 100 => 1
[2,1,1,1] => [2,3,4,1] => [1,2,4,3] => 001 => 1
[1,1,1,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,1,3,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,3,1,1] => [1,3,4,2] => [2,4,1,3] => 100 => 1
[3,1,1,1] => [2,3,4,1] => [1,2,4,3] => 001 => 1
[1,1,1,4] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,1,4,1] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,4,1,1] => [1,3,4,2] => [2,4,1,3] => 100 => 1
[4,1,1,1] => [2,3,4,1] => [1,2,4,3] => 001 => 1
[1,1,2,2] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,1,2] => [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,2,2,1] => [1,4,2,3] => [3,4,1,2] => 100 => 1
[2,1,1,2] => [2,3,1,4] => [1,3,2,4] => 010 => 1
[2,1,2,1] => [2,4,1,3] => [1,3,4,2] => 010 => 1
[2,2,1,1] => [3,4,1,2] => [3,1,4,2] => 110 => 2
[1,1,2,3] => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,1,3,2] => [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,2,1,3] => [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,2,3,1] => [1,4,2,3] => [3,4,1,2] => 100 => 1
[1,3,1,2] => [1,3,4,2] => [2,4,1,3] => 100 => 1
[1,3,2,1] => [1,4,3,2] => [4,3,1,2] => 101 => 2
[2,1,1,3] => [2,3,1,4] => [1,3,2,4] => 010 => 1
[2,1,3,1] => [2,4,1,3] => [1,3,4,2] => 010 => 1
[2,3,1,1] => [3,4,1,2] => [3,1,4,2] => 110 => 2
[3,1,1,2] => [2,3,4,1] => [1,2,4,3] => 001 => 1
[3,1,2,1] => [2,4,3,1] => [4,1,3,2] => 110 => 2
[3,2,1,1] => [3,4,2,1] => [1,4,3,2] => 011 => 2

Description

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

Matching statistic: St000354

Mapping

Mp00055: Parking functions —to labelling permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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.

The following 5 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. St001624The breadth of a lattice. St000761The number of ascents in an integer composition. St001330The hat guessing number of a graph. St001946The number of descents in a parking function.