- FindStat

Your data matches 40 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000996
(load all 4 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000374

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [] => [] => 0
[1,0,1,0]
 => [1,2] => [1] => [1] => 0
[1,1,0,0]
 => [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,4,5,7,2,3,8,6] => [1,4,5,7,2,3,6] => [1,5,6,2,3,7,4] => ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,6,3,4,5,8,7] => [2,1,6,3,4,5,7] => [2,1,4,5,6,3,7] => ? = 2
[1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [2,1,6,3,4,8,5,7] => [2,1,6,3,4,5,7] => [2,1,4,5,6,3,7] => ? = 2
[1,1,0,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,1,7,8,3,4,5,6] => [2,1,7,3,4,5,6] => [2,1,4,5,6,7,3] => ? = 2
[1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,3,8,5,6,7] => [2,4,1,3,5,6,7] => [3,1,4,2,5,6,7] => ? = 2
[1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,0]
 => [2,5,7,1,8,3,4,6] => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => ? = 3
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [2,5,7,8,1,3,4,6] => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => ? = 3
[1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [2,6,1,3,4,5,7,8] => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 2
[1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => [2,6,1,7,3,8,4,5] => [2,6,1,7,3,4,5] => [3,1,5,6,7,2,4] => ? = 3
[1,1,0,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => [2,6,7,1,3,4,8,5] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 3
[1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [2,6,7,8,1,3,4,5] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 3
[1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,7,1,3,4,5,6,8] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [2,7,1,8,3,4,5,6] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,7,8,1,3,4,5,6] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 2
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => [4,1,6,2,8,3,5,7] => [4,1,6,2,3,5,7] => [2,4,5,1,6,3,7] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
 => [4,6,1,2,3,5,7,8] => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 2
[1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
 => [4,6,1,2,3,7,5,8] => [4,6,1,2,3,7,5] => [3,4,5,1,7,2,6] => ? = 3
[1,1,1,1,0,1,1,0,0,1,1,0,0,0,0,0]
 => [4,6,1,8,2,3,5,7] => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => [5,1,6,2,3,4,7,8] => [5,1,6,2,3,4,7] => [2,4,5,6,1,3,7] => ? = 2
[1,1,1,1,1,0,0,1,0,0,0,1,1,0,0,0]
 => [5,1,6,2,3,8,4,7] => [5,1,6,2,3,4,7] => [2,4,5,6,1,3,7] => ? = 2
[1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => [5,6,1,2,3,7,8,4] => [5,6,1,2,3,7,4] => [3,4,5,7,1,2,6] => ? = 3
[1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
 => [5,6,1,2,3,8,4,7] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 2
[1,1,1,1,1,0,1,0,0,0,1,1,0,0,0,0]
 => [5,6,1,2,8,3,4,7] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => [5,6,7,1,2,3,4,8] => [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 3
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 3
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => [5,6,8,1,2,3,4,7] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [1,8,9,2,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [8,9,1,2,3,4,5,6,7,10] => [8,9,1,2,3,4,5,6,7] => [3,4,5,6,7,8,9,1,2] => ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [1,8,2,9,3,4,5,6,7] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [1,7,9,2,3,4,5,6,8] => [1,7,2,3,4,5,6,8] => [1,3,4,5,6,7,2,8] => ? = 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [1,9,10,2,3,4,5,6,7,8] => [1,9,2,3,4,5,6,7,8] => [1,3,4,5,6,7,8,9,2] => ? = 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [2,8,1,3,4,5,6,7,9] => [2,8,1,3,4,5,6,7] => [3,1,4,5,6,7,8,2] => ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [1,8,2,3,4,5,6,9,7] => [1,8,2,3,4,5,6,7] => [1,3,4,5,6,7,8,2] => ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [6,7,8,9,1,2,3,4,5] => [6,7,8,1,2,3,4,5] => [4,5,6,7,8,1,2,3] => ? = 3
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,5,8,6,9,7] => [1,2,3,4,5,8,6,7] => [1,2,3,4,5,7,8,6] => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,4,2,5,3,7,6,9,8] => [1,4,2,5,3,7,6,8] => [1,3,5,2,4,7,6,8] => ? = 3
[1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [2,9,10,1,3,4,5,6,7,8] => [2,9,1,3,4,5,6,7,8] => [3,1,4,5,6,7,8,9,2] => ? = 2
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
 => [8,9,10,1,2,3,4,5,6,7] => [8,9,1,2,3,4,5,6,7] => [3,4,5,6,7,8,9,1,2] => ? = 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
 => [6,7,8,9,10,1,2,3,4,5] => [6,7,8,9,1,2,3,4,5] => [5,6,7,8,9,1,2,3,4] => ? = 4
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
 => [7,8,9,10,1,2,3,4,5,6] => [7,8,9,1,2,3,4,5,6] => [4,5,6,7,8,9,1,2,3] => ? = 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
(load all 4 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

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: St000318

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 88%●distinct values known / distinct values provided: 60%

Values

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

Description

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

Matching statistic: St000337

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [] => [] => 0
[1,0,1,0]
 => [1,2] => [1] => [1] => 0
[1,1,0,0]
 => [2,1] => [1] => [1] => 0
[1,0,1,0,1,0]
 => [1,2,3] => [1,2] => [1,2] => 0
[1,0,1,1,0,0]
 => [1,3,2] => [1,2] => [1,2] => 0
[1,1,0,0,1,0]
 => [2,1,3] => [2,1] => [2,1] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,1] => [2,1] => 1
[1,1,1,0,0,0]
 => [3,1,2] => [1,2] => [1,2] => 0
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,3,2] => [1,3,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => [1,2,3] => [1,2,3] => 0
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,3] => [2,1,3] => 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => [3,1,2] => [2,3,1] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => [3,1,2] => [2,3,1] => 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => [1,3,4,2] => [1,4,2,3] => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => [2,4,1,3] => [3,1,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,2,3,4,6,7,8,5] => [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,2,3,4,7,5,8,6] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,2,3,4,7,8,5,6] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3,5,4,8,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,2,3,5,6,7,8,4] => [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,3,6,4,5,7,8] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,2,3,6,4,7,5,8] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,2,3,6,4,7,8,5] => [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,2,3,6,4,8,5,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,2,3,6,7,4,8,5] => [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,7,4,5,6,8] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,2,3,7,4,5,8,6] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,2,3,7,4,8,5,6] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,2,4,3,6,7,8,5] => [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,8,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,2,4,5,6,7,8,3] => [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => ? = 4
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,2,4,6,7,8,3,5] => [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,2,5,3,6,4,7,8] => [1,2,5,3,6,4,7] => [1,2,4,6,3,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,2,5,3,6,7,4,8] => [1,2,5,3,6,7,4] => [1,2,4,7,3,5,6] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,2,5,3,6,7,8,4] => [1,2,5,3,6,7,4] => [1,2,4,7,3,5,6] => ? = 3
[1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,2,5,3,8,4,6,7] => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => [1,2,5,6,3,4,7,8] => [1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,2,5,6,3,7,8,4] => [1,2,5,6,3,7,4] => [1,2,5,7,3,4,6] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,2,5,6,7,3,8,4] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => ? = 3
[1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,2,5,6,7,8,3,4] => [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,2,6,3,4,7,8,5] => [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,2,6,3,4,8,5,7] => [1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,2,6,3,7,4,5,8] => [1,2,6,3,7,4,5] => [1,2,4,6,7,3,5] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,2,7,3,4,5,8,6] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,3,2,4,5,7,6,8] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
 => [1,3,2,4,6,7,8,5] => [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,5,4,6,7,8] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,7,6,8] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,5,4,8,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,2,5,8,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,6,4,5,7,8] => [1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
 => [1,3,2,6,4,7,5,8] => [1,3,2,6,4,7,5] => [1,3,2,5,7,4,6] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,3,2,6,4,8,5,7] => [1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,3,2,7,4,5,6,8] => [1,3,2,7,4,5,6] => [1,3,2,5,6,7,4] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => [1,3,2,7,4,8,5,6] => [1,3,2,7,4,5,6] => [1,3,2,5,6,7,4] => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,6,7,5,8] => [1,3,4,2,6,7,5] => [1,4,2,3,7,5,6] => ? = 4
[1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,3,4,2,6,7,8,5] => [1,3,4,2,6,7,5] => [1,4,2,3,7,5,6] => ? = 4
[1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,4,2,8,5,6,7] => [1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => ? = 2
[1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,3,5,2,6,7,4,8] => [1,3,5,2,6,7,4] => [1,4,2,7,3,5,6] => ? = 4
[1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
 => [1,3,5,2,6,8,4,7] => [1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => ? = 3
[1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [1,3,5,2,7,4,8,6] => [1,3,5,2,7,4,6] => [1,4,2,6,3,7,5] => ? = 3
[1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [1,3,5,8,2,4,6,7] => [1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => ? = 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: St000159

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 84%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => [1,0]
 => []
 => ?
 => ? = 0
[1,0,1,0]
 => [1,1,0,0]
 => []
 => ?
 => ? = 0
[1,1,0,0]
 => [1,0,1,0]
 => [1]
 => []
 => 0
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [2]
 => []
 => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1]
 => [1]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [2,1]
 => [1]
 => 1
[1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => [1]
 => []
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => [3]
 => []
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [2,2]
 => [2]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [3,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [2]
 => []
 => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => [1,1]
 => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => [2,1]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => [2,1]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1]
 => [1]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1]
 => [1]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [2,1]
 => [1]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1]
 => []
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => []
 => ?
 => ? = 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4]
 => []
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => [3]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => [3]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [3]
 => []
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => [2,2]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => [2,2]
 => 1
[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]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => [3,2]
 => 2
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => [2,2]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => [2]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => [2]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => [2]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2]
 => []
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => [3,1,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => [3,1,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => [1,1,1]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => [2,2,1]
 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => [2,2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => [3,2,1]
 => 3
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => [3,2,1]
 => 3
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => [2,2,1]
 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => [2,1,1]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => [2,1,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => [2,1,1]
 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => [1,1]
 => 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => [3,1]
 => 2
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => [3,1]
 => 2
[1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => [1,1]
 => 1
[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
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0
[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]
 => [5,4,2,2]
 => ? = 3
[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]
 => [5,4,2,2]
 => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2]
 => [5,3,3,2]
 => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2]
 => [5,3,3,2]
 => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2]
 => [4,4,3,2]
 => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2]
 => [4,4,3,2]
 => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2]
 => [5,4,3,2]
 => ? = 4
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2]
 => [5,4,3,2]
 => ? = 4
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2]
 => [4,4,3,2]
 => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,1,1]
 => [5,3,3,1,1]
 => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,1,1]
 => [5,3,3,1,1]
 => ? = 3
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,1,1]
 => [4,4,3,1,1]
 => ? = 3
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,1,1]
 => [4,4,3,1,1]
 => ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1,1]
 => [5,4,3,1,1]
 => ? = 4
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1,1]
 => [5,4,3,1,1]
 => ? = 4
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => [4,4,3,1,1]
 => ? = 3
[1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [4,4,4,2,2,1]
 => [4,4,2,2,1]
 => ? = 3
[1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,2,1]
 => [4,4,2,2,1]
 => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,2,1]
 => [5,4,2,2,1]
 => ? = 4
[1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,2,1]
 => [5,4,2,2,1]
 => ? = 4
[1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => [4,4,2,2,1]
 => ? = 3
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,5,3,3,2,1]
 => [5,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,3,2,1]
 => [5,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,4,4,3,2,1]
 => [4,4,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,4,3,2,1]
 => [4,4,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,2,1]
 => [5,4,3,2,1]
 => ? = 5
[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]
 => [6,5,4,3,2,1]
 => [5,4,3,2,1]
 => ? = 5
[1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,2,1]
 => [4,4,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [4,4,3,3,2,1]
 => [4,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => [4,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [5,4,3,3,2,1]
 => [4,3,3,2,1]
 => ? = 4
[1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => [5,3,2,2,1]
 => ? = 4
[1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => [5,3,2,2,1]
 => ? = 4
[1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,1,1]
 => [5,4,2,1,1]
 => ? = 4
[1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,1]
 => [5,4,2,1,1]
 => ? = 4
[1,1,1,0,0,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
 => [5,5,4,3,1]
 => [5,4,3,1]
 => ? = 4
[1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,1]
 => [5,4,3,1]
 => ? = 4
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => []
 => ?
 => ? = 0
[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]
 => [6,6,4,4]
 => ?
 => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,1,0,0]
 => [6,4,4,4]
 => ?
 => ? = 1
[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]
 => [7,6,5,4]
 => [6,5,4]
 => ? = 3
[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]
 => [5,5,5,3,3]
 => ?
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [7,6,5,3,3]
 => ?
 => ? = 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: St001124

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 79%●distinct values known / distinct values provided: 50%

Values

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

Matching statistic: St000245

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 90%

Values

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

Description

The number of ascents of a permutation.

Matching statistic: St001036

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001036: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 76%●distinct values known / distinct values provided: 70%

Values

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

Description

The number of inner corners of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000672

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 75%●distinct values known / distinct values provided: 90%

Values

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

Description

The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$ for some

$(r,a,b)$ . This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

The following 30 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000021The number of descents of a permutation. St000204The number of internal nodes of a binary tree. St000155The number of exceedances (also excedences) of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000316The number of non-left-to-right-maxima of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000619The number of cyclic descents of a permutation. St000711The number of big exceedences of a permutation. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000710The number of big deficiencies of a permutation. St000390The number of runs of ones in a binary word. St000742The number of big ascents of a permutation after prepending zero. St001864The number of excedances of a signed permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000646The number of big ascents 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. St001960The number of descents of a permutation minus one if its first entry is not one.