- FindStat

Your data matches 63 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000993

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St001232

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St000021
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000354
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of recoils of a permutation. A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$. In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.

Matching statistic: St000541
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000541: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. For a permutation $\pi$ of length $n$, this is the number of indices $2 \leq j \leq n$ such that for all $1 \leq i < j$, the pair $(i,j)$ is an inversion of $\pi$.

Matching statistic: St000619

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of cyclic descents of a permutation. For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.

Matching statistic: St000831

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000831: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of indices that are either descents or recoils. This is, for a permutation $\pi$ of length $n$, this statistics counts the set $$\{ 1 \leq i < n : \pi(i) > \pi(i+1) \text{ or } \pi^{-1}(i) > \pi^{-1}(i+1)\}.$$

Matching statistic: St001061

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The number of indices that are both descents and recoils of a permutation.

Matching statistic: St001489
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].

Matching statistic: St000325
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%

Values

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

Description

The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.

The following 53 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001490The number of connected components of a skew partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. 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 $eA$. St001557The number of inversions of the second entry of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001668The number of points of the poset minus the width of the poset. St000738The first entry in the last row of a standard tableau. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000153The number of adjacent cycles of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. St000074The number of special entries. St000141The maximum drop size of a permutation. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St001684The reduced word complexity of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001569The maximal modular displacement of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000682The Grundy value of Welter's game on a binary word. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000662The staircase size of the code of a permutation. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001838The number of nonempty primitive factors of a binary word. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000019The cardinality of the support of a permutation. St000214The number of adjacencies of a permutation. St000054The first entry of the permutation. St000507The number of ascents of a standard tableau. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000883The number of longest increasing subsequences of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000733The row containing the largest entry of a standard tableau. St000157The number of descents of a standard tableau.