- FindStat

Your data matches 62 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000996
(load all 15 compositions to match this statistic)

Mapping

St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive left-to-right maxima of a permutation. This is the number of left-to-right maxima that are not right-to-left minima.

Matching statistic: St000340

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 73%

Values

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

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

Matching statistic: St000159

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 69%●distinct values known / distinct values provided: 45%

Values

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

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St000318

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 55% ●values known / values provided: 66%●distinct values known / distinct values provided: 55%

Values

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

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000374
(load all 17 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exclusive right-to-left minima of a permutation. This is the number of right-to-left minima that are not left-to-right maxima. This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3. Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there do not exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000703

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of deficiencies of a permutation. This is defined as

$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$ The number of exceedances is [[St000155]].

Matching statistic: St000337

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%

Values

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

Description

The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. For a permutation

$\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines

$\textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$ where

$\textrm{inv} \, \tau_{i}$ is the number of inversions of

$\tau_{i}$ .

Matching statistic: St000925

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of topologically connected components of a set partition. For example, the set partition

$\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components

$\{1,4,5,6\}$ and

$\{2,3\}$ . The number of set partitions with only one block is [[oeis:A099947]].

Matching statistic: St000662

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 91%

Values

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

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

Matching statistic: St001124

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 45% ●values known / values provided: 52%●distinct values known / distinct values provided: 45%

Values

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

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

The following 52 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000052The number of valleys of a Dyck path not on the x-axis. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000105The number of blocks in the set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000167The number of leaves of an ordered tree. St000912The number of maximal antichains in a poset. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000292The number of ascents of a binary word. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000647The number of big descents of a permutation. St000354The number of recoils of a permutation. St000021The number of descents of a permutation. St000542The number of left-to-right-minima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St000740The last entry of a permutation. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St000083The number of left oriented leafs of a binary tree except the first one. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001346The number of parking functions that give the same permutation. St001330The hat guessing number of a graph. St000991The number of right-to-left minima of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000702The number of weak deficiencies of a permutation. St001712The number of natural descents of a standard Young tableau. 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

$eA$ .