Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000230
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000230: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [2,1] => {{1,2}}
 => 1
[1,2] => [1,0,1,0]
 => [3,1,2] => {{1,3},{2}}
 => 3
[2,1] => [1,1,0,0]
 => [2,3,1] => {{1,2,3}}
 => 1
[1,2,3] => [1,0,1,0,1,0]
 => [4,1,2,3] => {{1,4},{2},{3}}
 => 6
[1,3,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => {{1,3,4},{2}}
 => 3
[2,1,3] => [1,1,0,0,1,0]
 => [2,4,1,3] => {{1,2,4},{3}}
 => 4
[2,3,1] => [1,1,0,1,0,0]
 => [4,3,1,2] => {{1,4},{2,3}}
 => 3
[3,1,2] => [1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 1
[3,2,1] => [1,1,1,0,0,0]
 => [2,3,4,1] => {{1,2,3,4}}
 => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
 => 10
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => {{1,4,5},{2},{3}}
 => 6
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => {{1,3,5},{2},{4}}
 => 7
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => {{1,5},{2},{3,4}}
 => 6
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => {{1,3,4,5},{2}}
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => {{1,2,5},{3},{4}}
 => 8
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => {{1,2,4,5},{3}}
 => 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => {{1,5},{2,3},{4}}
 => 7
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => {{1,5},{2,4},{3}}
 => 6
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => {{1,4,5},{2,3}}
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => {{1,4,5},{2,3}}
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 5
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => {{1,2,5},{3,4}}
 => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => {{1,2,3,5},{4}}
 => 5
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => {{1,2,5},{3,4}}
 => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => {{1,5},{2,3,4}}
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => {{1,2,3,4,5}}
 => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => {{1,6},{2},{3},{4},{5}}
 => 15
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => {{1,5,6},{2},{3},{4}}
 => 10
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => {{1,4,6},{2},{3},{5}}
 => 11
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
 => 10
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
 => 6
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
 => 6
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => {{1,3,6},{2},{4},{5}}
 => 12
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
 => 7
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}}
 => 11
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
 => 10
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
 => 6
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
 => 6
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
 => 8
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
 => 7
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
 => 8
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
 => 7
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
 => 6
Description
Sum of the minimal elements of the blocks of a set partition.
Matching statistic: St000756
(load all 29 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000756: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [2,1] => 1
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3
[2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 6
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 3
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 4
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 3
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 10
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 6
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 7
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 6
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 8
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 7
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 6
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 5
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 5
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => 15
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,5,1] => 10
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,4,6,1] => 11
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,4,1] => 10
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,5,4,1] => 6
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,5,4,1] => 6
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,3,5,6,1] => 12
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,3,6,5,1] => 7
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,3,6,1] => 11
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,3,1] => 10
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,5,3,1] => 6
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,5,3,1] => 6
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => 8
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,4,6,3,1] => 7
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [2,5,4,3,6,1] => 8
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,4,6,3,1] => 7
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,4,3,1] => 6
Description
The sum of the positions of the left to right maxima of a permutation. The generating function for this statistic is $$\sum_{\pi\in\mathfrak S_n} q^{slrmax(pi)} = \prod_{k=1}^n (q^k+k-1),$$ see [prop. 2.6., 1].
Matching statistic: St000947
(load all 2 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000947: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3
[2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 6
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 4
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 10
[1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 6
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 7
[1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 6
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 3
[2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 8
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4
[2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 7
[2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 6
[2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 3
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 5
[3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 4
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 15
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 10
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => 11
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 10
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 6
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 6
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => 12
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => 7
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => 11
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 10
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 6
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 6
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 8
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 7
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 8
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 7
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => 6
[] => []
 => [1,0]
 => [1,0]
 => ? = 1
Description
The major index east count of a Dyck path. The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]]. The '''major index east count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = E\}$.