- FindStat

Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000996

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000028

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of stack-sorts needed to sort a permutation. A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series. Let $W_t(n,k)$ be the number of permutations of size $n$ with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$ are symmetric and unimodal. We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted. Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).

Matching statistic: St000155

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 38% ●values known / values provided: 38%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of exceedances (also excedences) of a permutation. This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$. It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].

Matching statistic: St001683

Mapping

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.

Matching statistic: St001685

Mapping

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.

Matching statistic: St000355

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00153: Standard tableaux —inverse promotion⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.

Matching statistic: St001744

Mapping

Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$ such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$. Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation. An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows. Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.

Matching statistic: St001435

Mapping

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of missing boxes in the first row.