- FindStat

Your data matches 32 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000117
(load all 23 compositions to match this statistic)

Mapping

St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of centered tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.

Matching statistic: St000022
(load all 72 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000221
(load all 167 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong fixed points of a permutation.

$i$ is called a strong fixed point of

$\pi$ if 1.

$j < i$ implies

$\pi_j < \pi_i$ , and 2.

$j > i$ implies

$\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

Matching statistic: St000241
(load all 31 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclical small excedances. A cyclical small excedance is an index

$i$ such that

$\pi_i = i+1$ considered cyclically.

Matching statistic: St000315
(load all 13 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of isolated vertices of a graph.

Matching statistic: St000475
(load all 7 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000895
(load all 22 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ones on the main diagonal of an alternating sign matrix.

Matching statistic: St001008
(load all 16 compositions to match this statistic)

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001008: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000674
(load all 29 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.

Matching statistic: St000164

Mapping

Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00116: Perfect matchings —Kasraoui-Zeng⟶ Perfect matchings
St000164: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of short pairs. A short pair is a matching pair of the form

$(i,i+1)$ .

The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000215The number of adjacencies of a permutation, zero appended. St000445The number of rises of length 1 of a Dyck path. St001479The number of bridges of a graph. St000011The number of touch points (or returns) of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St001826The maximal number of leaves on a vertex of a graph. St001672The restrained domination number of a graph. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000894The trace of an alternating sign matrix. St000907The number of maximal antichains of minimal length in a poset. St000237The number of small exceedances. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000239The number of small weak excedances. 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. St001545The second Elser number of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001903The number of fixed points of a parking function.