- FindStat

Your data matches 373 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000667

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The greatest common divisor of the parts of the partition.

Matching statistic: St000655
(load all 2 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].

Matching statistic: St001075

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 75% ●values known / values provided: 85%●distinct values known / distinct values provided: 75%

Values

[1,2] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[2,1] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,3] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[3,1] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,4] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[2,3] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[3,2] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[4,1] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[1,5] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[2,4] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1
[4,2] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[5,1] => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 1
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1,2,3,4,5},{6}}
 => 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1
[1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4},{5},{6,7,8}}
 => ? = 1
[1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 1
[1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8}}
 => ? = 1
[1,1,1,1,1,1,1,1,2] => [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => {{1,2,3,4,5,6,7,8},{9}}
 => ? = 1
[1,1,1,1,1,1,1,2,1] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6,7},{8},{9}}
 => ? = 1
[1,1,1,1,1,1,2,1,1] => [6,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5,6},{7},{8,9}}
 => ? = 1
[1,1,1,1,1,1,3,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => {{1,2,3,4,5,6},{7},{8}}
 => ? = 1
[1,1,1,1,1,2,1,1,1] => [5,1,3] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4,5},{6},{7,8,9}}
 => ? = 1
[1,1,1,1,1,2,1,2] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => {{1,2,3,4,5},{6},{7},{8}}
 => ? = 1
[1,1,1,1,1,2,2,1] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => {{1,2,3,4,5},{6,7},{8}}
 => ? = 1
[1,1,1,1,1,3,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => {{1,2,3,4,5},{6},{7,8}}
 => ? = 1
[1,1,1,1,2,1,1,1,1] => [4,1,4] => [1,1,1,1,0,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5},{6,7,8,9}}
 => ? = 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => {{1,2,3,4},{5},{6,7},{8}}
 => ? = 1
[1,1,1,1,2,1,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => {{1,2,3,4},{5},{6},{7},{8}}
 => ? = 1
[1,1,1,1,2,2,1,1] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6},{7,8}}
 => ? = 2
[1,1,1,1,3,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => {{1,2,3,4},{5},{6,7,8}}
 => ? = 1
[1,1,1,2,1,1,1,1,1] => [3,1,5] => [1,1,1,0,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8,9}}
 => ? = 1
[1,1,1,2,1,1,1,2] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => {{1,2,3},{4},{5,6,7},{8}}
 => ? = 1
[1,1,1,2,1,1,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => {{1,2,3},{4},{5,6},{7},{8}}
 => ? = 1
[1,1,1,2,1,2,1,1] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => {{1,2,3},{4},{5},{6},{7,8}}
 => ? = 1
[1,1,1,2,2,1,1,1] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => {{1,2,3},{4,5},{6,7,8}}
 => ? = 1
[1,1,1,3,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1,2,3},{4},{5,6,7,8}}
 => ? = 1
[1,1,2,1,1,1,1,1,1] => [2,1,6] => [1,1,0,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8,9}}
 => ? = 1
[1,1,2,1,1,1,1,2] => [2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => {{1,2},{3},{4,5,6,7},{8}}
 => ? = 1
[1,1,2,1,1,1,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => {{1,2},{3},{4,5,6},{7},{8}}
 => ? = 1
[1,1,2,1,1,2,1,1] => [2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5},{6},{7,8}}
 => ? = 1
[1,1,2,1,2,1,1,1] => [2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2},{3},{4},{5},{6,7,8}}
 => ? = 1
[1,1,2,2,1,1,1,1] => [2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4},{5,6,7,8}}
 => ? = 2
[1,1,3,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1,2},{3},{4,5,6,7,8}}
 => ? = 1
[1,2,1,1,1,1,1,1,1] => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => {{1},{2},{3,4,5,6,7,8,9}}
 => ? = 1
[1,2,1,1,1,1,1,2] => [1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => {{1},{2},{3,4,5,6,7},{8}}
 => ? = 1
[1,2,1,1,1,1,2,1] => [1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => {{1},{2},{3,4,5,6},{7},{8}}
 => ? = 1
[1,2,1,1,1,2,1,1] => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => {{1},{2},{3,4,5},{6},{7,8}}
 => ? = 1
[1,2,1,1,2,1,1,1] => [1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4},{5},{6,7,8}}
 => ? = 1
[1,2,1,2,1,1,1,1] => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2},{3},{4},{5,6,7,8}}
 => ? = 1
[1,2,2,1,1,1,1,1] => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2,3},{4,5,6,7,8}}
 => ? = 1
[2,1,1,1,1,1,1,1,1] => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => {{1},{2,3,4,5,6,7,8,9}}
 => ? = 1
[2,1,1,1,1,1,2,1] => [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => {{1},{2,3,4,5,6},{7},{8}}
 => ? = 1
[2,1,1,1,1,2,1,1] => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => {{1},{2,3,4,5},{6},{7,8}}
 => ? = 1
[2,1,1,1,2,1,1,1] => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => {{1},{2,3,4},{5},{6,7,8}}
 => ? = 1
[2,1,1,2,1,1,1,1] => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3},{4},{5,6,7,8}}
 => ? = 1
[2,1,2,1,1,1,1,1] => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => {{1},{2},{3},{4,5,6,7,8}}
 => ? = 1
[1,1,1,1,1,1,1,1,2,2] => [8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => {{1,2,3,4,5,6,7,8},{9,10}}
 => ? = 2
[1,1,1,1,1,1,2,2,1,1] => [6,2,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0]
 => ?
 => ? = 2
[1,1,1,1,1,1,2,1,1,2] => [6,1,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,1,0,0,1,0]
 => {{1,2,3,4,5,6},{7},{8,9},{10}}
 => ? = 1
[1,1,1,1,2,2,1,1,1,1] => [4,2,4] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6},{7,8,9,10}}
 => ? = 2
[1,1,1,1,2,2,2,2] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6,7,8}}
 => ? = 4
[1,1,1,1,2,1,1,2,1,1] => [4,1,2,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => ?
 => ? = 1

Description

The minimal size of a block of a set partition.

Matching statistic: St000685
(load all 3 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%

Values

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

Description

The dominant dimension of the LNakayama algebra associated to a Dyck path. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].

Matching statistic: St000487
(load all 3 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000487: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 75%

Values

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

Description

The length of the shortest cycle of a permutation.

Matching statistic: St000210
(load all 3 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 75%

Values

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

Description

Minimum over maximum difference of elements in cycles. Given a cycle

$C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is

$\max \{ a_i-a_j | a_i, a_j \in C \}$ . The statistic is then the minimum of this value over all cycles in the permutation. For example, all permutations with a fixed-point has statistic value 0, and all permutations of

$[n]$ with only one cycle, has statistic value

$n-1$ .

Matching statistic: St000260
(load all 13 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 64%●distinct values known / distinct values provided: 25%

Values

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

Description

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

Matching statistic: St001236
(load all 13 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St001236: Integer compositions ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 75%

Values

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

Description

The dominant dimension of the corresponding Comp-Nakayama algebra.

Matching statistic: St001481
(load all 13 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 75%

Values

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

Description

The minimal height of a peak of a Dyck path.

Matching statistic: St000455
(load all 41 compositions to match this statistic)

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 43%●distinct values known / distinct values provided: 25%

Values

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

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

The following 363 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000900The minimal number of repetitions of a part in an integer composition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000782The indicator function of whether a given perfect matching is an L & P matching. St001568The smallest positive integer that does not appear twice in the partition. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000264The girth of a graph, which is not a tree. St000259The diameter of a connected graph. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000007The number of saliances of the permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St000546The number of global descents of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000842The breadth of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000570The Edelman-Greene number of a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001307The number of induced stars on four vertices in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000862The number of parts of the shifted shape of a permutation. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000286The number of connected components of the complement of a graph. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000223The number of nestings in the permutation. St000534The number of 2-rises of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000237The number of small exceedances. St000648The number of 2-excedences of a permutation. St000731The number of double exceedences of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000234The number of global ascents of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000662The staircase size of the code of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000456The monochromatic index of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000374The number of exclusive right-to-left minima of a permutation. St000647The number of big descents of a permutation. St001665The number of pure excedances of a permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000451The length of the longest pattern of the form k 1 2. St000750The number of occurrences of the pattern 4213 in a permutation. St001381The fertility of a permutation. St001394The genus of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001728The number of invisible descents of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000255The number of reduced Kogan faces with the permutation as type. St001344The neighbouring number of a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001964The interval resolution global dimension of a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001715The number of non-records in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000633The size of the automorphism group of a poset. St000717The number of ordinal summands of a poset. St001399The distinguishing number of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of 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. St000908The length of the shortest maximal antichain in a poset. St001396Number of triples of incomparable elements in a finite poset. St000028The number of stack-sorts needed to sort a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001490The number of connected components of a skew partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001208The 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

$K[x]/(x^n)$ . St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001487The number of inner corners of a skew partition. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000096The number of spanning trees of a graph. St000221The number of strong fixed points of a permutation. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000542The number of left-to-right-minima of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001577The minimal number of edges to add or remove to make a graph a cograph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001734The lettericity of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. 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

$eA$ . St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St000023The number of inner peaks of a permutation. St000037The sign of a permutation. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000155The number of exceedances (also excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000353The number of inner valleys of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000627The exponent of a binary word. St000646The number of big ascents of a permutation. St000654The first descent of a permutation. St000711The number of big exceedences of a permutation. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000756The sum of the positions of the left to right maxima of a permutation. St000779The tier of a permutation. St000843The decomposition number of a perfect matching. St000876The number of factors in the Catalan decomposition of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000991The number of right-to-left minima of a permutation. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001048The number of leaves in the subtree containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001260The permanent of an alternating sign matrix. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001470The cyclic holeyness of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001828The Euler characteristic of a graph. St000039The number of crossings of a permutation. St000042The number of crossings of a perfect matching. St000051The size of the left subtree of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000133The "bounce" of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000241The number of cyclical small excedances. St000296The length of the symmetric border of a binary word. St000308The height of the tree associated to a permutation. St000317The cycle descent number of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000461The rix statistic of a permutation. St000485The length of the longest cycle of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000638The number of up-down runs of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000674The number of hills of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000822The Hadwiger number of the graph. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001082The number of boxed occurrences of 123 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of

$Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra

$A$ such that

$eA$ is a minimal faithful projective-injective module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001371The length of the longest Yamanouchi prefix of a binary word. St001429The number of negative entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001471The magnitude of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001517The length of a longest pair of twins in a permutation. St001519The pinnacle sum of a permutation. St001520The number of strict 3-descents. St001530The depth of a Dyck path. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001555The order of a signed permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000670The reversal length of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St001388The number of non-attacking neighbors of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St000545The number of parabolic double cosets with minimal element being the given permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both.