- FindStat

Your data matches 151 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000021
(load all 8 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000023
(load all 14 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of inner peaks of a permutation. The number of peaks including the boundary is [[St000092]].

Matching statistic: St000035
(load all 11 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left outer peaks of a permutation. A left outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ . In other words, it is a peak in the word

$[0,w_1,..., w_n]$ . This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

Matching statistic: St000884
(load all 7 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of isolated descents of a permutation. A descent

$i$ is isolated if neither

$i+1$ nor

$i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].

Matching statistic: St001729
(load all 3 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of visible descents of a permutation. A visible descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i+1) \leq \min(i, \pi(i))$ .

Matching statistic: St001928
(load all 7 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence

$\pi_i,\pi_{i+1},\dots,\pi_k$ contributes

$\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.

Matching statistic: St000099
(load all 14 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].

Matching statistic: St000325
(load all 8 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => 2 = 1 + 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 8 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 0 + 1
[1,2] => [1,2] => 1 = 0 + 1
[2,1] => [1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => 1 = 0 + 1
[1,3,2] => [1,3,2] => 2 = 1 + 1
[2,1,3] => [1,3,2] => 2 = 1 + 1
[2,3,1] => [1,2,3] => 1 = 0 + 1
[3,1,2] => [1,2,3] => 1 = 0 + 1
[3,2,1] => [1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,4,3,2] => [1,4,2,3] => 2 = 1 + 1
[2,1,3,4] => [1,3,4,2] => 2 = 1 + 1
[2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[2,3,1,4] => [1,4,2,3] => 2 = 1 + 1
[2,3,4,1] => [1,2,3,4] => 1 = 0 + 1
[2,4,1,3] => [1,3,2,4] => 2 = 1 + 1
[2,4,3,1] => [1,2,4,3] => 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => 2 = 1 + 1
[3,1,4,2] => [1,4,2,3] => 2 = 1 + 1
[3,2,1,4] => [1,4,2,3] => 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => 1 = 0 + 1
[4,1,3,2] => [1,3,2,4] => 2 = 1 + 1
[4,2,1,3] => [1,3,2,4] => 2 = 1 + 1
[4,2,3,1] => [1,2,3,4] => 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,2,4,5,3] => [1,2,4,5,3] => 2 = 1 + 1
[1,2,5,3,4] => [1,2,5,3,4] => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => 3 = 2 + 1
[1,3,4,2,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,3,4,5,2] => [1,3,4,5,2] => 2 = 1 + 1
[1,3,5,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,3,5,4,2] => [1,3,5,2,4] => 2 = 1 + 1
[1,4,2,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,2,5,3] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,2,5] => [1,4,2,5,3] => 3 = 2 + 1
[1,4,3,5,2] => [1,4,2,3,5] => 2 = 1 + 1
[1,4,5,2,3] => [1,4,5,2,3] => 2 = 1 + 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 4 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ 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] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 1
[1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 1
[2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,4,3,2,5] => [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => 1

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]].

The following 141 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001214The aft of an integer partition. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000054The first entry of the permutation. St000092The number of outer peaks of a permutation. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000024The number of double up and double down steps of a Dyck path. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000168The number of internal nodes of an ordered tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000360The number of occurrences of the pattern 32-1. St000362The size of a minimal vertex cover of a graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000387The matching number of a graph. St000536The pathwidth of a graph. St000647The number of big descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001277The degeneracy of a graph. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001689The number of celebrities in a graph. St001712The number of natural descents of a standard Young tableau. St001737The number of descents of type 2 in a permutation. St001792The arboricity of a graph. St001812The biclique partition number of a graph. St000010The length of the partition. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000288The number of ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000443The number of long tunnels of a Dyck path. St000507The number of ascents of a standard tableau. St000522The number of 1-protected nodes of a rooted tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000439The position of the first down step of a Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000619The number of cyclic descents of a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000919The number of maximal left branches of a binary tree. St000568The hook number of a binary tree. St000083The number of left oriented leafs of a binary tree except the first one. St000291The number of descents of a binary word. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000702The number of weak deficiencies of a permutation. St000946The sum of the skew hook positions in a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000806The semiperimeter of the associated bargraph. St001427The number of descents of a signed permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001487The number of inner corners of a skew partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001060The distinguishing index of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001624The breadth of a lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001935The number of ascents in a parking function. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation.