- FindStat

Your data matches 19 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001085

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern

$213$ , where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.

Matching statistic: St000758
(load all 7 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
St000758: Integer compositions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 = 0 + 1
[1,1] => [1,1] => 1 = 0 + 1
[2] => [2] => 1 = 0 + 1
[1,1,1] => [1,1,1] => 1 = 0 + 1
[1,2] => [2,1] => 1 = 0 + 1
[2,1] => [1,2] => 2 = 1 + 1
[3] => [3] => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1] => 1 = 0 + 1
[1,1,2] => [2,1,1] => 1 = 0 + 1
[1,2,1] => [1,2,1] => 2 = 1 + 1
[1,3] => [3,1] => 1 = 0 + 1
[2,1,1] => [1,1,2] => 2 = 1 + 1
[2,2] => [2,2] => 2 = 1 + 1
[3,1] => [1,3] => 2 = 1 + 1
[4] => [4] => 1 = 0 + 1
[1,1,1,1,1] => [1,1,1,1,1] => 1 = 0 + 1
[1,1,1,2] => [2,1,1,1] => 1 = 0 + 1
[1,1,2,1] => [1,2,1,1] => 2 = 1 + 1
[1,1,3] => [3,1,1] => 1 = 0 + 1
[1,2,1,1] => [1,1,2,1] => 2 = 1 + 1
[1,2,2] => [2,2,1] => 2 = 1 + 1
[1,3,1] => [1,3,1] => 2 = 1 + 1
[1,4] => [4,1] => 1 = 0 + 1
[2,1,1,1] => [1,1,1,2] => 2 = 1 + 1
[2,1,2] => [2,1,2] => 2 = 1 + 1
[2,2,1] => [1,2,2] => 2 = 1 + 1
[2,3] => [3,2] => 2 = 1 + 1
[3,1,1] => [1,1,3] => 2 = 1 + 1
[3,2] => [2,3] => 2 = 1 + 1
[4,1] => [1,4] => 2 = 1 + 1
[5] => [5] => 1 = 0 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,2] => [2,1,1,1,1] => 1 = 0 + 1
[1,1,1,2,1] => [1,2,1,1,1] => 2 = 1 + 1
[1,1,1,3] => [3,1,1,1] => 1 = 0 + 1
[1,1,2,1,1] => [1,1,2,1,1] => 2 = 1 + 1
[1,1,2,2] => [2,2,1,1] => 2 = 1 + 1
[1,1,3,1] => [1,3,1,1] => 2 = 1 + 1
[1,1,4] => [4,1,1] => 1 = 0 + 1
[1,2,1,1,1] => [1,1,1,2,1] => 2 = 1 + 1
[1,2,1,2] => [2,1,2,1] => 2 = 1 + 1
[1,2,2,1] => [1,2,2,1] => 2 = 1 + 1
[1,2,3] => [3,2,1] => 2 = 1 + 1
[1,3,1,1] => [1,1,3,1] => 2 = 1 + 1
[1,3,2] => [2,3,1] => 2 = 1 + 1
[1,4,1] => [1,4,1] => 2 = 1 + 1
[1,5] => [5,1] => 1 = 0 + 1
[2,1,1,1,1] => [1,1,1,1,2] => 2 = 1 + 1
[2,1,1,2] => [2,1,1,2] => 2 = 1 + 1
[2,1,2,1] => [1,2,1,2] => 2 = 1 + 1
[1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1] => ? = 0 + 1
[1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1] => ? = 0 + 1
[1,1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1,1] => ? = 1 + 1
[2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2] => ? = 1 + 1
[2,2,2,2,2] => [2,2,2,2,2] => ? = 1 + 1
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,3] => ? = 1 + 1
[10] => [10] => ? = 0 + 1
[2,2,2,2,2,2] => [2,2,2,2,2,2] => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1] => ? = 0 + 1

Description

The length of the longest staircase fitting into an integer composition. For a given composition

$c_1,\dots,c_n$ , this is the maximal number

$\ell$ such that there are indices

$i_1 < \dots < i_\ell$ with

$c_{i_k} \geq k$ , see [def.3.1, 1]

Matching statistic: St000862
(load all 7 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1,0]
 => [1] => 1 = 0 + 1
[1,1] => [1,1] => [1,0,1,0]
 => [1,2] => 1 = 0 + 1
[2] => [2] => [1,1,0,0]
 => [2,1] => 1 = 0 + 1
[1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 0 + 1
[1,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1 = 0 + 1
[2,1] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[3] => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 1 = 0 + 1
[1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1 = 0 + 1
[1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1 = 0 + 1
[1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2 = 1 + 1
[1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 1 = 0 + 1
[2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 1 + 1
[2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 1 + 1
[3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2 = 1 + 1
[4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1 = 0 + 1
[1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 1 = 0 + 1
[1,1,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 1 = 0 + 1
[1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 2 = 1 + 1
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 1 = 0 + 1
[1,2,1,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 2 = 1 + 1
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2 = 1 + 1
[1,3,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 2 = 1 + 1
[1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 1 = 0 + 1
[2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 2 = 1 + 1
[2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2 = 1 + 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2 = 1 + 1
[2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2 = 1 + 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2 = 1 + 1
[3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2 = 1 + 1
[4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 2 = 1 + 1
[5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1 = 0 + 1
[1,1,1,1,1,1] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => 1 = 0 + 1
[1,1,1,1,2] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,3,4,5,6] => 1 = 0 + 1
[1,1,1,2,1] => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6] => 2 = 1 + 1
[1,1,1,3] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,2,1,4,5,6] => 1 = 0 + 1
[1,1,2,1,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => 2 = 1 + 1
[1,1,2,2] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => 2 = 1 + 1
[1,1,3,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,4,3,2,5,6] => 2 = 1 + 1
[1,1,4] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,3,2,1,5,6] => 1 = 0 + 1
[1,2,1,1,1] => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => 2 = 1 + 1
[1,2,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,1,3,5,4,6] => 2 = 1 + 1
[1,2,2,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,6] => 2 = 1 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1,5,4,6] => 2 = 1 + 1
[1,3,1,1] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,5,4,3,6] => 2 = 1 + 1
[1,3,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,1,5,4,3,6] => 2 = 1 + 1
[1,4,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,5,4,3,2,6] => 2 = 1 + 1
[1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 1 = 0 + 1
[2,1,1,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => 2 = 1 + 1
[2,1,1,2] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,6,5] => 2 = 1 + 1
[2,1,2,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,5] => 2 = 1 + 1
[2,1,1,1,2] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [2,1,3,4,5,7,6] => ? = 1 + 1
[1,1,1,1,2,1,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7,8] => ? = 1 + 1
[1,1,2,1,1,2] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,3,4,6,5,7,8] => ? = 1 + 1
[1,1,3,1,1,1] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,2,3,6,5,4,7,8] => ? = 1 + 1
[1,3,1,1,1,1] => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,4,7,6,5,8] => ? = 1 + 1
[1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => [3,2,1,4,7,6,5,8] => ? = 1 + 1
[2,2,3,1] => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,4,3,2,6,5,8,7] => ? = 1 + 1
[2,3,2,1] => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2,6,5,4,8,7] => ? = 2 + 1
[2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,6,5,4,8,7] => ? = 1 + 1
[3,1,1,1,1,1] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,8,7,6] => ? = 1 + 1
[3,2,2,1] => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,5,4,8,7,6] => ? = 2 + 1
[3,2,3] => [3,2,3] => [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
[1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7,8,9] => ? = 1 + 1
[1,1,1,1,1,2,1,1] => [1,1,2,1,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,4,3,5,6,7,8,9] => ? = 1 + 1
[1,3,1,1,1,1,1] => [1,1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,2,3,4,5,8,7,6,9] => ? = 1 + 1
[3,1,1,1,1,1,1] => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,6,9,8,7] => ? = 1 + 1
[1,1,1,1,1,1,1,2,1] => [1,2,1,1,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,2,4,5,6,7,8,9,10] => ? = 1 + 1
[3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,4,5,6,7,10,9,8] => ? = 1 + 1
[2,2,2,2,2,2] => [2,2,2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1] => ?
 => ? => ? = 0 + 1

Description

The number of parts of the shifted shape of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the number of parts of the shifted shape.

Matching statistic: St000619

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclic descents of a permutation. For a permutation

$\pi$ of

$\{1,\ldots,n\}$ , this is given by the number of indices

$1 \leq i \leq n$ such that

$\pi(i) > \pi(i+1)$ where we set

$\pi(n+1) = \pi(1)$ .

Matching statistic: St000662
(load all 3 compositions to match this statistic)

Mapping

Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 67%

Values

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

Description

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

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

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

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

$(c_1,\ldots,c_n)$ with

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

$k$ such that there exists a subsequence

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

$c(\pi)$ with

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

Matching statistic: St000455
(load all 5 compositions to match this statistic)

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

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

Description

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

Matching statistic: St001597

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001597: Skew partitions ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 67%

Values

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

Description

The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.

Matching statistic: St000470

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.

Matching statistic: St001394

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001394: Permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%

Values

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

Description

The genus of a permutation. The genus

$g(\pi)$ of a permutation

$\pi\in\mathfrak S_n$ is defined via the relation

$n+1-2g(\pi) = z(\pi) + z(\pi^{-1} \zeta ),$ where

$\zeta = (1,2,\dots,n)$ is the long cycle and

$z(\cdot)$ is the number of cycles in the permutation.

Matching statistic: St000021

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 67%

Values

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

Description

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

The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000325The width of the tree associated to a permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St000456The monochromatic index of a connected graph. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001960The number of descents of a permutation minus one if its first entry is not one. St001487The number of inner corners of a skew partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation.