- FindStat

Your data matches 66 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001195
(load all 115 compositions to match this statistic)

Mapping

St001195: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 1
[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]
 => 1
[1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => 1
[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]
 => 1
[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]
 => 1
[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]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => 1
[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]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => 0
[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]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => 1
[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]
 => 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]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => 0

Description

The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ .

Matching statistic: St001174

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001174: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ .

Matching statistic: St001273

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001273: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 0

Description

The projective dimension of the first term in an injective coresolution of the regular module. The algebra has the double centraliser property when 0 is returned and it is 1-Gorenstein in case a number < =1 is returned.

Matching statistic: St000007
(load all 3 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000920
(load all 2 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 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]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 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]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 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]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 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]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 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]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 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]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 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]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 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]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 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]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1

Description

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Matching statistic: St001257
(load all 3 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001257: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1

Description

The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J.

Matching statistic: St000260
(load all 35 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%

Values

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

Description

The radius of a connected graph. This is the minimum eccentricity of any vertex.

Matching statistic: St001199
(load all 58 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00003: Alternating sign matrices —rotate counterclockwise⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => ? = 0
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => ? = 0
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => 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]]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 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]]
 => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0
[1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => 1
[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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[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]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[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]]
 => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [[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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => [[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [[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]]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[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]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [[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]]
 => [[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]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1

Description

The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Matching statistic: St001933

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%

Values

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

Description

The largest multiplicity of a part in an integer partition.

Matching statistic: St001198
(load all 63 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00003: Alternating sign matrices —rotate counterclockwise⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 66%●distinct values known / distinct values provided: 50%

Values

[1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [[0,0,1],[0,1,0],[1,0,0]]
 => [1,1,1,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [[0,1,0],[0,0,1],[1,0,0]]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [1,1,1,0,0,0]
 => ? = 0 + 1
[1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,1,0,1,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,0,1,1,0,0]
 => 2 = 1 + 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]]
 => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 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]]
 => [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 0 + 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]]
 => [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[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]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [[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]]
 => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 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]]
 => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 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]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[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],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [[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]]
 => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [[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]]
 => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,0,1,-1,1],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,0,0,0,1],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[0,1,0,-1,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[0,1,-1,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,1,0,0,0],[0,0,0,0,1],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [[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]]
 => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[0,0,1,0,0],[1,0,-1,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[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]]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [[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]]
 => [[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]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 2 = 1 + 1

Description

The 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

$eA$ .

The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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

$eA$ . 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. St000264The girth of a graph, which is not a tree. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000307The number of rowmotion orbits of a poset. St000456The monochromatic index of a connected graph. St000259The diameter of a connected graph. St001545The second Elser number of a connected graph. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001964The interval resolution global dimension of a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001060The distinguishing index of a graph. St000647The number of big descents of a permutation. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. 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. St001383The BG-rank of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. 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. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000237The number of small exceedances. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001372The length of a longest cyclic run of ones of a binary word. St000925The number of topologically connected components of a set partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation.