- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000007

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000989

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 64% ●values known / values provided: 79%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of final rises of a permutation. For a permutation

$\pi$ of length

$n$ , this is the maximal

$k$ such that

$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$ Equivalently, this is

$n-1$ minus the position of the last descent [[St000653]].

Matching statistic: St001232
(load all 3 compositions to match this statistic)

Mapping

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 64%

Values

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

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Matching statistic: St001879
(load all 3 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 45%

Values

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

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Matching statistic: St001880
(load all 3 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 45%

Values

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

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St001557

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 36%

Values

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

Description

The number of inversions of the second entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.