- FindStat

Your data matches 170 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000445
(load all 29 compositions to match this statistic)

Mapping

St000445: 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]
 => 2
[1,1,0,0]
 => 0
[1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => 3
[1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 3
[1,0,1,1,0,1,0,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => 0

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St001126
(load all 13 compositions to match this statistic)

Mapping

St001126: 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]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 1
[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]
 => 1
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => 2
[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]
 => 0
[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]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => 3

Description

Number of simple module that are 1-regular in the corresponding Nakayama algebra.

Matching statistic: St000022
(load all 27 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing 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]
 => [1,2] => 2
[1,1,0,0]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => 3
[1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => 0
[1,1,1,0,0,0]
 => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 0

Description

The number of fixed points of a permutation.

Matching statistic: St000215
(load all 15 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000215: 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]
 => [1,2] => 0
[1,1,0,0]
 => [2,1] => 2
[1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 3

Description

The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form

$(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic,

$\operatorname{adj_0}$ , counts adjacencies in the word with a zero appended.

$(\operatorname{adj_0}, \operatorname{des})$ and

$(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].

Matching statistic: St000241
(load all 11 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000241: 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]
 => [1,2] => 0
[1,1,0,0]
 => [2,1] => 2
[1,0,1,0,1,0]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => 3
[1,1,1,0,0,0]
 => [3,2,1] => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 4
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 3
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 5
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 3
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 3
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 3

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: St000895
(load all 11 compositions to match this statistic)

Mapping

Mp00137: Dyck paths —to symmetric ASM⟶ 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
[1,0,1,0]
 => [[1,0],[0,1]]
 => 2
[1,1,0,0]
 => [[0,1],[1,0]]
 => 0
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => 0
[1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => 2
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => 2
[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
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => 2
[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
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => 0
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => 1
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => 0
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 2
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => 2
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [[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
[1,0,1,1,0,0,1,1,0,0]
 => [[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
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[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
[1,1,0,0,1,0,1,1,0,0]
 => [[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
[1,1,0,0,1,1,0,0,1,0]
 => [[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
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => 0

Description

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

Matching statistic: St001189
(load all 7 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001189: 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
[1,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 3

Description

The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000221
(load all 6 compositions to match this statistic)

Mapping

Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key 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
[1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 2
[1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 0
[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 3
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 1
[1,1,1,0,0,0]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 2
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 0
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 2
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 2
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 0
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,4,3,2] => 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 3
[1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
 => [[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]]
 => [1,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
 => [[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]]
 => [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 3
[1,0,1,1,0,1,1,0,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
 => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[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]]
 => [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
 => [[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]]
 => [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [[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]]
 => [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1
[1,1,0,0,1,1,1,0,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [2,1,5,4,3] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => [1,4,3,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
 => [1,2,5,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,5,4,3,2] => 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: St000475
(load all 2 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing 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]
 => [1,2] => [1,1]
 => 2
[1,1,0,0]
 => [2,1] => [2]
 => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,1,1]
 => 3
[1,0,1,1,0,0]
 => [1,3,2] => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1]
 => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3]
 => 0
[1,1,1,0,0,0]
 => [3,2,1] => [2,1]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,1,1,1]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [2,1,1]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [3,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,1]
 => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,1]
 => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,2]
 => 0
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [3,1]
 => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [4]
 => 0
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [2,1,1]
 => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,1]
 => 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [2,1,1]
 => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [2,2]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [3,1,1]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => [2,1,1,1]
 => 3
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [2,1,1,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [2,2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [3,1,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [4,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => [3,1,1]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => [2,1,1,1]
 => 3
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => [3,1,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => [2,1,1,1]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => [2,2,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,1,1]
 => 3
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,2,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,2,1]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [3,2]
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => [2,2,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [3,1,1]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [3,2]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [4,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [5]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => [4,1]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => [3,1,1]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => [4,1]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,5,3,4,1] => [3,1,1]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => [3,2]
 => 0

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000502
(load all 3 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,1,0,0]
 => {{1,2}}
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 0
[1,1,0,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 2
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 0
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 3
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => 0
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => {{1,4,5},{2,3}}
 => 2
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 0
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => {{1,2,5},{3},{4}}
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => {{1,2,4,5},{3}}
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => {{1,2,5},{3,4}}
 => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => {{1,2,3,5},{4}}
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 4
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => {{1,6},{2},{3},{4},{5}}
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2},{3},{4}}
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => {{1,6},{2},{3},{4,5}}
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => {{1,4,6},{2},{3},{5}}
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2},{3}}
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => {{1,6},{2},{3,4},{5}}
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => {{1,5,6},{2},{3,4}}
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => {{1,6},{2},{3,5},{4}}
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => {{1,3,6},{2},{4},{5}}
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => {{1,3,5,6},{2},{4}}
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => {{1,6},{2},{3,4,5}}
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => {{1,3,6},{2},{4,5}}
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => {{1,3,4,6},{2},{5}}
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => {{1,3,4,5,6},{2}}
 => 3
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => {{1,6},{2,3},{4},{5}}
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => {{1,5,6},{2,3},{4}}
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => {{1,6},{2,3},{4,5}}
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => {{1,4,6},{2,3},{5}}
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => {{1,4,5,6},{2,3}}
 => 3
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => {{1,6},{2,4},{3},{5}}
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => {{1,5,6},{2,4},{3}}
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => {{1,6},{2,5},{3},{4}}
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => {{1,2,6},{3},{4},{5}}
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => {{1,2,5,6},{3},{4}}
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => {{1,6},{2,4,5},{3}}
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => {{1,2,6},{3},{4,5}}
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => {{1,2,4,6},{3},{5}}
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => {{1,2,4,5,6},{3}}
 => 3

Description

The number of successions of a set partitions. This is the number of indices

$i$ such that

$i$ and

$i+1$ belonging to the same block.

The following 160 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000932The number of occurrences of the pattern UDU in a Dyck path. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St000992The alternating sum of the parts of an integer partition. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000248The number of anti-singletons of a set partition. St000117The number of centered tunnels of a Dyck path. St000148The number of odd parts of a partition. St000315The number of isolated vertices of a graph. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001657The number of twos in an integer partition. St001691The number of kings in a graph. St000247The number of singleton blocks of a set partition. St000504The cardinality of the first block of a set partition. St000925The number of topologically connected components of a set partition. St000674The number of hills of a Dyck path. St000441The number of successions of a permutation. St001484The number of singletons of an integer partition. St000894The trace of an alternating sign matrix. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000443The number of long tunnels of a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001461The number of topologically connected components of the chord diagram of a permutation. St000237The number of small exceedances. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001571The Cartan determinant of the integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000444The length of the maximal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000654The first descent of a permutation. St000873The aix statistic of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. 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. St000313The number of degree 2 vertices of a graph. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St000239The number of small weak excedances. St001061The number of indices that are both descents and recoils of a permutation. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000931The number of occurrences of the pattern UUU in a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. 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

$eA$ . St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001118The acyclic chromatic index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St000137The Grundy value of an integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000474Dyson's crank of a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001587Half of the largest even part of an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000145The Dyson rank of a partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000939The number of characters of the symmetric group whose value on the partition is positive. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001569The maximal modular displacement of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001060The distinguishing index of a graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000731The number of double exceedences of a permutation. St000365The number of double ascents of a permutation. St001405The number of bonds in a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000366The number of double descents of a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000284The Plancherel distribution on integer partitions. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001903The number of fixed points of a parking function. St001948The number of augmented double ascents of a permutation. St000264The girth of a graph, which is not a tree. St000317The cycle descent number of a permutation. St000732The number of double deficiencies of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001516The number of cyclic bonds of a permutation. St001180Number of indecomposable injective modules with projective dimension at most 1. St001875The number of simple modules with projective dimension at most 1. St000454The largest eigenvalue of a graph if it is integral.