- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000648
(load all 7 compositions to match this statistic)

Mapping

Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000648: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of 2-excedences of a permutation. This is the number of positions

$1\leq i\leq n$ such that

$\sigma(i)=i+2$ .

Matching statistic: St001465
(load all 6 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacent transpositions in the cycle decomposition of a permutation.

Matching statistic: St000502

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of successions of a set partitions. This is the number of indices

$i$ such that

$i$ and

$i+1$ belonging to the same block.

Matching statistic: St000237
(load all 3 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of small exceedances. This is the number of indices

$i$ such that

$\pi_i=i+1$ .

Matching statistic: St001061
(load all 6 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St000214
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of adjacencies of a permutation. An adjacency of a permutation

$\pi$ is an index

$i$ such that

$\pi(i)-1 = \pi(i+1)$ . Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].

Matching statistic: St001230
(load all 8 compositions to match this statistic)

Mapping

St001230: Dyck paths ⟶ ℤResult quality: 59% ●values known / values provided: 59%●distinct values known / distinct values provided: 75%

Values

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

Description

The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.

Matching statistic: St000534
(load all 2 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000534: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of 2-rises of a permutation. A 2-rise of a permutation

$\pi$ is an index

$i$ such that

$\pi(i)+2 = \pi(i+1)$ . For 1-rises, or successions, see [[St000441]].

Matching statistic: St000264

Mapping

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 25%

Values

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

Description

The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.

Matching statistic: St001621

Mapping

Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%

Values

[1,0]
 => [1,0]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1,0,0]
 => ([],2)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
 => ? = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
 => ? = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
 => ? = 0 + 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
 => ? = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([],5)
 => ?
 => ? = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16),(16,1)],17)
 => ? = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,7),(2,8),(2,9),(3,9),(3,11),(3,12),(4,8),(4,10),(4,12),(5,7),(5,10),(5,11),(7,13),(7,14),(8,13),(8,15),(9,14),(9,15),(10,13),(10,16),(11,14),(11,16),(12,15),(12,16),(13,17),(14,17),(15,17),(16,1),(16,17),(17,6)],18)
 => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4)],5)
 => ([(0,1),(1,2),(1,3),(1,4),(1,5),(2,9),(2,10),(2,11),(3,7),(3,8),(3,11),(4,6),(4,8),(4,10),(5,6),(5,7),(5,9),(6,12),(6,15),(7,12),(7,13),(8,12),(8,14),(9,13),(9,15),(10,14),(10,15),(11,13),(11,14),(12,16),(13,16),(14,16),(15,16)],17)
 => ? = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,2),(0,3),(0,4),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,1),(6,9),(7,9),(8,9),(9,5)],10)
 => ? = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,11),(2,12),(2,13),(3,9),(3,10),(3,13),(4,8),(4,10),(4,12),(5,8),(5,9),(5,11),(6,16),(7,16),(8,1),(8,17),(8,18),(9,14),(9,17),(10,15),(10,17),(11,14),(11,18),(12,15),(12,18),(13,14),(13,15),(14,19),(15,19),(17,6),(17,19),(18,7),(18,19),(19,16)],20)
 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,10),(7,10),(8,2),(8,10),(9,1),(10,9)],11)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(3,9),(4,7),(4,9),(5,7),(5,8),(7,10),(8,10),(9,10),(10,1),(10,2)],11)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([(0,5),(2,7),(2,8),(3,6),(3,8),(4,6),(4,7),(5,2),(5,3),(5,4),(6,9),(7,9),(8,9),(9,1)],10)
 => ? = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(3,10),(4,6),(4,10),(5,6),(5,7),(6,11),(7,11),(8,9),(10,2),(10,11),(11,1),(11,8)],12)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([(0,5),(1,8),(2,7),(2,9),(3,6),(3,9),(4,6),(4,7),(5,2),(5,3),(5,4),(6,10),(7,10),(9,1),(9,10),(10,8)],11)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ([(0,4),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,5),(5,1),(5,2),(5,3),(6,9),(7,9),(8,9)],10)
 => ? = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(3,4)],5)
 => ?
 => ? = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(2,9),(2,10),(3,6),(3,8),(4,6),(4,7),(5,2),(5,7),(5,8),(6,11),(7,9),(7,11),(8,10),(8,11),(9,12),(10,12),(11,12),(12,1)],13)
 => ? = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,8),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(7,13),(9,12),(10,6),(10,12),(11,7),(11,12),(12,1),(12,13),(13,8)],14)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,5),(1,9),(1,10),(2,6),(2,8),(3,6),(3,7),(4,1),(4,7),(4,8),(5,2),(5,3),(5,4),(6,12),(7,9),(7,12),(8,10),(8,12),(9,11),(10,11),(12,11)],13)
 => ? = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(2,3),(2,4)],5)
 => ([(0,1),(0,2),(0,3),(1,11),(1,13),(2,11),(2,12),(3,4),(3,5),(3,12),(3,13),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,15),(6,17),(7,15),(7,18),(8,16),(8,17),(9,16),(9,18),(10,15),(10,16),(11,14),(12,6),(12,7),(12,14),(13,8),(13,9),(13,14),(14,17),(14,18),(15,19),(16,19),(17,19),(18,19)],20)
 => ? = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
 => ? = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,8),(2,6),(2,7),(3,10),(3,11),(4,9),(4,11),(5,9),(5,10),(6,12),(7,12),(8,12),(9,13),(10,13),(11,1),(11,2),(11,13),(13,7),(13,8)],14)
 => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
 => ? = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ([(0,4),(0,5),(1,8),(2,7),(2,9),(3,7),(3,10),(4,6),(5,2),(5,3),(5,6),(6,9),(6,10),(7,11),(9,11),(10,1),(10,11),(11,8)],12)
 => ? = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,3),(0,4),(4,1),(4,2)],5)
 => ([(0,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,2),(4,3),(4,6),(5,1),(5,4),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
 => ? = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
 => ? = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,6),(1,7),(2,8),(2,10),(3,9),(3,11),(4,9),(4,12),(5,2),(5,11),(5,12),(6,14),(7,14),(8,13),(9,15),(10,6),(10,13),(11,8),(11,15),(12,1),(12,10),(12,15),(13,14),(15,7),(15,13)],16)
 => ? = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,1),(0,2),(1,11),(2,4),(2,5),(2,11),(3,6),(3,7),(4,8),(4,10),(5,8),(5,9),(6,13),(7,13),(8,12),(9,6),(9,12),(10,7),(10,12),(11,3),(11,9),(11,10),(12,13)],14)
 => ? = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
 => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 0 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ([(0,4),(0,5),(1,7),(1,9),(2,7),(2,8),(3,6),(4,10),(5,3),(5,10),(6,8),(6,9),(7,11),(8,11),(9,11),(10,1),(10,2),(10,6)],12)
 => ? = 0 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
 => ? = 0 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(3,2),(4,1),(4,3)],5)
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 0 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,3),(3,4),(4,1),(4,2)],5)
 => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ?
 => ? = 0 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,6),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,7),(4,9),(4,11),(5,7),(5,8),(5,10),(6,2),(6,3),(6,4),(6,5),(7,13),(7,16),(8,13),(8,14),(9,13),(9,15),(10,14),(10,16),(11,15),(11,16),(12,14),(12,15),(13,17),(14,17),(15,17),(16,17),(17,1)],18)
 => ? = 0 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([(0,2),(0,3),(0,4),(2,8),(2,9),(3,7),(3,9),(4,7),(4,8),(5,1),(6,5),(7,10),(8,10),(9,10),(10,6)],11)
 => ? = 0 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => ([(0,5),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(2,13),(3,4),(3,5),(3,6),(3,13),(4,9),(4,11),(4,12),(5,8),(5,10),(5,12),(6,7),(6,10),(6,11),(7,14),(7,15),(8,14),(8,16),(9,15),(9,16),(10,14),(10,17),(11,15),(11,17),(12,16),(12,17),(13,7),(13,8),(13,9),(14,18),(15,18),(16,18),(17,18),(18,1)],19)
 => ? = 0 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(0,4),(0,5),(0,6),(2,10),(3,10),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(7,11),(8,11),(9,11),(10,1),(11,2),(11,3)],12)
 => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 1 = 0 + 1

Description

The number of atoms of a lattice. An element of a lattice is an '''atom''' if it covers the least element.

The following 1 statistic also match your data. Click on any of them to see the details.

St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.