- FindStat

Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001525

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of symmetric hooks on the diagonal of a partition.

Matching statistic: St001195

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

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

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: St001208

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ? = 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 0
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ? = 0
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ? = 0
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 0
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 0
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 0
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ? = 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ? = 0
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ? = 0
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ? = 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 0
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ? = 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,1,2,5,3,7,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => ? = 0
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => ? = 0
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => ? = 0
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => ? = 0
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => ? = 0
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => ? = 0
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => ? = 0
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => ? = 0
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1

Description

The number of connected components of the quiver of

$A/T$ when

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

$A$ of

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

Matching statistic: St000508

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000508: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 0 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 0 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 0 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 0 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => ? = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => ? = 0 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => ? = 0 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => ? = 0 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 0 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,7,8,10],[2,4,6,9,11,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,3,5,6,8,11],[2,4,7,9,10,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[1,3,5,6,8,10],[2,4,7,9,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [[1,3,5,6,8,9],[2,4,7,10,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,3,5,6,7,11],[2,4,8,9,10,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,3,5,6,7,10],[2,4,8,9,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[1,3,5,6,7,9],[2,4,8,10,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,3,5,6,7,8],[2,4,9,10,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,3,4,7,9,11],[2,5,6,8,10,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,3,4,7,9,10],[2,5,6,8,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,3,4,7,8,11],[2,5,6,9,10,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[1,3,4,7,8,10],[2,5,6,9,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,3,4,7,8,9],[2,5,6,10,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[1,3,4,6,9,11],[2,5,7,8,10,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[1,3,4,6,9,10],[2,5,7,8,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[1,3,4,6,8,11],[2,5,7,9,10,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[1,3,4,6,8,10],[2,5,7,9,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [[1,3,4,6,8,9],[2,5,7,10,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[1,3,4,6,7,11],[2,5,8,9,10,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[1,3,4,6,7,10],[2,5,8,9,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [[1,3,4,6,7,9],[2,5,8,10,11,12]]
 => ? = 0 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1

Description

Eigenvalues of the random-to-random operator acting on a simple module. The simple module of the symmetric group indexed by a partition

$\lambda$ has dimension equal to the number of standard tableaux of shape

$\lambda$ . Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape

$\lambda$ ; this statistic gives all the eigenvalues of the operator acting on the module [1].

Matching statistic: St001001

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

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

Description

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001371

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 0 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? = 0 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? = 0 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? = 0 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? = 0 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 0 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 101011100010 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 101100101010 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 101100101100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 101100110010 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 101100110100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 101100111000 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 101101001010 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 101101001100 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 101101010010 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 101101010100 => ? = 0 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 101101011000 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 101101100010 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 101101100100 => ? = 0 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 101101101000 => ? = 0 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1

Description

The length of the longest Yamanouchi prefix of a binary word. This is the largest index

$i$ such that in each of the prefixes

$w_1$ ,

$w_1w_2$ ,

$w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.

Matching statistic: St001557

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 0 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 0 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 0 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 0 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 0 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ? = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ? = 0 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ? = 0 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ? = 0 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 0 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,1,2,3,7,4,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [7,1,2,3,6,4,5] => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [7,1,2,5,3,4,6] => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [7,1,2,6,3,4,5] => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [6,1,2,5,3,7,4] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [7,1,2,5,6,3,4] => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [7,1,4,2,3,5,6] => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [7,1,5,2,3,4,6] => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [6,1,7,2,3,4,5] => ? = 0 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [6,1,5,2,3,7,4] => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => ? = 0 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [7,1,5,2,6,3,4] => ? = 0 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0 = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 0 = 1 - 1

Description

The number of inversions of the second entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

Matching statistic: St001730

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => ? = 0 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => ? = 0 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => ? = 0 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => ? = 0 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => ? = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => ? = 0 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => ? = 0 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => ? = 0 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => ? = 0 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 101010101010 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 101010101100 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 101010110010 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 101010110100 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 101010111000 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 101011001010 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 101011001100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 101011010010 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 101011010100 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 101011011000 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 101011100010 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 101011100100 => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 101011101000 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => 101011110000 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => 101100101010 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => 101100101100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 101100110010 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 101100110100 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 101100111000 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 101101001010 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 101101001100 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => 101101010010 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 101101010100 => ? = 0 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => 101101011000 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 101101100010 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 101101100100 => ? = 0 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => 101101101000 => ? = 0 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 10101100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 10101010 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 10110010 => 0 = 1 - 1

Description

The number of times the path corresponding to a binary word crosses the base line. Interpret each

$0$ as a step

$(1,-1)$ and

$1$ as a step

$(1,1)$ . Then this statistic counts the number of times the path crosses the

$x$ -axis.

Matching statistic: St001803

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => ? = 0 - 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 0 - 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 0 - 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 - 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 0 - 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 0 - 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => ? = 0 - 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => ? = 0 - 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => ? = 0 - 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => ? = 0 - 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 0 - 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => ? = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => ? = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,7,8,10],[2,4,6,9,11,12]]
 => ? = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,3,5,6,8,11],[2,4,7,9,10,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[1,3,5,6,8,10],[2,4,7,9,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [[1,3,5,6,8,9],[2,4,7,10,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,3,5,6,7,11],[2,4,8,9,10,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,3,5,6,7,10],[2,4,8,9,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[1,3,5,6,7,9],[2,4,8,10,11,12]]
 => ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,3,5,6,7,8],[2,4,9,10,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,3,4,7,9,11],[2,5,6,8,10,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,3,4,7,9,10],[2,5,6,8,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,3,4,7,8,11],[2,5,6,9,10,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[1,3,4,7,8,10],[2,5,6,9,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,3,4,7,8,9],[2,5,6,10,11,12]]
 => ? = 1 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[1,3,4,6,9,11],[2,5,7,8,10,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[1,3,4,6,9,10],[2,5,7,8,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[1,3,4,6,8,11],[2,5,7,9,10,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[1,3,4,6,8,10],[2,5,7,9,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [[1,3,4,6,8,9],[2,5,7,10,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[1,3,4,6,7,11],[2,5,8,9,10,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[1,3,4,6,7,10],[2,5,8,9,11,12]]
 => ? = 0 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [[1,3,4,6,7,9],[2,5,8,10,11,12]]
 => ? = 0 - 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 0 = 1 - 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 0 = 1 - 1

Description

The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau

$T$ is the skew row-strict tableau obtained by gluing two copies of

$T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals

$\max_C\big(2\ell(T) - \ell(C)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals

$0$ , if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.

Matching statistic: St001804

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 20%

Values

[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => ? = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => ? = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => ? = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => ? = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [4,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [5,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,7,8,11],[2,4,6,9,10,12]]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [4,3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,7,8,10],[2,4,6,9,11,12]]
 => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [3,3,3,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,7,8,9],[2,4,6,10,11,12]]
 => ? = 2 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[1,3,5,6,9,11],[2,4,7,8,10,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[1,3,5,6,9,10],[2,4,7,8,11,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [[1,3,5,6,8,11],[2,4,7,9,10,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [[1,3,5,6,8,10],[2,4,7,9,11,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [[1,3,5,6,8,9],[2,4,7,10,11,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [[1,3,5,6,7,11],[2,4,8,9,10,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,3,5,6,7,10],[2,4,8,9,11,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [[1,3,5,6,7,9],[2,4,8,10,11,12]]
 => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [[1,3,5,6,7,8],[2,4,9,10,11,12]]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [[1,3,4,7,9,11],[2,5,6,8,10,12]]
 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
 => [4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[1,3,4,7,9,10],[2,5,6,8,11,12]]
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
 => [5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[1,3,4,7,8,11],[2,5,6,9,10,12]]
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
 => [4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[1,3,4,7,8,10],[2,5,6,9,11,12]]
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[1,3,4,7,8,9],[2,5,6,10,11,12]]
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [[1,3,4,6,9,11],[2,5,7,8,10,12]]
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[1,3,4,6,9,10],[2,5,7,8,11,12]]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [[1,3,4,6,8,11],[2,5,7,9,10,12]]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [[1,3,4,6,8,10],[2,5,7,9,11,12]]
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [3,3,2,1,1]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [[1,3,4,6,8,9],[2,5,7,10,11,12]]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[1,3,4,6,7,11],[2,5,8,9,10,12]]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
 => [4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[1,3,4,6,7,10],[2,5,8,9,11,12]]
 => ? = 0 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
 => [3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [[1,3,4,6,7,9],[2,5,8,10,11,12]]
 => ? = 0 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0,0,0]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1

Description

The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. A cylindrical tableau associated with a standard Young tableau

$T$ is the skew row-strict tableau obtained by gluing two copies of

$T$ such that the inner shape is a rectangle. This statistic equals

$\max_C\big(\ell(C) - \ell(T)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.

The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001618The cardinality of the Frattini sublattice of a lattice.