- FindStat

Your data matches 140 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000838
(load all 7 compositions to match this statistic)

Mapping

St000838: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of terminal right-hand endpoints when the vertices are written in order. An opener (or left hand endpoint) of a perfect matching is a number that is matched with a larger number, which is then called a closer (or right hand endpoint). The opener-closer sequence of the perfect matching

$\{(1,3),(2,5),(4,6)\}$ is

$OOCOCC$ , so the number of terminal right-hand endpoints is

$2$ . The number of perfect matchings of

$\{1,\dots,2n\}$ with exactly

$T$ terminal closers, according to [1] computed in [2], is

$\frac{T(2n-T-1)!}{2^{n-T}(n-T)!}.$

Matching statistic: St001041

Mapping

St001041: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. The bijection between perfect matchings of

$\{1,\dots,2n\}$ and trees with

$n+1$ leaves is described in Example 5.2.6 of [1].

Matching statistic: St000025
(load all 16 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of initial rises of a Dyck path. In other words, this is the height of the first peak of

$D$ .

Matching statistic: St001184
(load all 32 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

Matching statistic: St000439
(load all 24 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The position of the first down step of a Dyck path.

Matching statistic: St000007
(load all 28 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000011
(load all 24 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

Matching statistic: St000054
(load all 22 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation

$\pi$ of

$n$ , together with its rotations, obtained by conjugating with the long cycle

$(1,\dots,n)$ . Drawing the labels

$1$ to

$n$ in this order on a circle, and the arcs

$(i, \pi(i))$ as straight lines, the rotation of

$\pi$ is obtained by replacing each number

$i$ by

$(i\bmod n) +1$ . Then,

$\pi(1)-1$ is the number of rotations of

$\pi$ where the arc

$(1, \pi(1))$ is a deficiency. In particular, if

$O(\pi)$ is the orbit of rotations of

$\pi$ , then the number of deficiencies of

$\pi$ equals

$\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).$

Matching statistic: St000066
(load all 9 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
St000066: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The column of the unique '1' in the first row of the alternating sign matrix. The generating function of this statistic is given by

$\binom{n+k-2}{k-1}\frac{(2n-k-1)!}{(n-k)!}\;\prod_{j=0}^{n-2}\frac{(3j+1)!}{(n+j)!},$ see [2].

Matching statistic: St000068
(load all 4 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in a poset.

The following 130 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000069The number of maximal elements of a poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000738The first entry in the last row of a standard tableau. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000326The position of the first one in a binary word after appending a 1 at the end. St000352The Elizalde-Pak rank of a permutation. St000383The last part of an integer composition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000617The number of global maxima of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000759The smallest missing part in an integer partition. St000843The decomposition number of a perfect matching. St000989The number of final rises of a permutation. St001050The number of terminal closers of a set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001481The minimal height of a peak of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000053The number of valleys of the Dyck path. St000061The number of nodes on the left branch of a binary tree. St000133The "bounce" of a permutation. St000203The number of external nodes of a binary tree. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000331The number of upper interactions of a Dyck path. St000504The cardinality of the first block of a set partition. St000546The number of global descents of a permutation. St000654The first descent of a permutation. St000734The last entry in the first row of a standard tableau. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000675The number of centered multitunnels of a Dyck path. St000717The number of ordinal summands of a poset. St000990The first ascent of a permutation. St000502The number of successions of a set partitions. St000993The multiplicity of the largest part of an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001568The smallest positive integer that does not appear twice in the partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001060The distinguishing index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000505The biggest entry in the block containing the 1. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000942The number of critical left to right maxima of the parking functions. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001115The number of even descents of a permutation. St000232The number of crossings of a set partition. St000422The energy of a graph, if it is integral. 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

$eA$ . St000451The length of the longest pattern of the form k 1 2. St000996The number of exclusive left-to-right maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001394The genus of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001812The biclique partition number of a graph. St000884The number of isolated descents of a permutation. St000035The number of left outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000647The number of big descents of a permutation. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000862The number of parts of the shifted shape of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001052The length of the exterior of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001801Half the number of preimage-image pairs of different parity in a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001427The number of descents of a signed permutation. St000028The number of stack-sorts needed to sort a permutation. St001114The number of odd descents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000461The rix statistic of a permutation. St000873The aix statistic of a permutation.