Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000444
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[5,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 3
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[[5,4,1],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001712
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 57%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2 = 3 - 1
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 3 - 1
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 0 = 1 - 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 1 = 2 - 1
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => 1 = 2 - 1
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 3 - 1
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2 = 3 - 1
[[3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 3 - 1
[[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 0 = 1 - 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 1 = 2 - 1
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => 1 = 2 - 1
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 3 - 1
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[5,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2 = 3 - 1
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 3 = 4 - 1
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 3 - 1
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2 = 3 - 1
[[5,4,1],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 4 - 1
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 1 = 2 - 1
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 3 - 1
[[5,4,2],[4,2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => ? = 3 - 1
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 3 - 1
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 1 = 2 - 1
[[4,4,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => [[1,3],[2,4]]
 => 1 = 2 - 1
[[4,3,2,1],[3,1,1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 4 - 1
[[5,4,2,1],[4,2,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,3,5,6,7,10],[2,4,8,9,11,12]]
 => ? = 4 - 1
[[5,4,3],[4,3]]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 2 - 1
[[4,4,3,1],[3,3,1]]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[1,2,3,5,8],[4,6,7,9,10]]
 => ? = 3 - 1
[[4,3,3,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[5,4,3,1],[4,3,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[1,3,4,5,7,10],[2,6,8,9,11,12]]
 => ? = 4 - 1
[[4,4,3,2],[3,3,2]]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[1,2,3,5,6],[4,7,8,9,10]]
 => ? = 2 - 1
[[4,3,3,2],[3,2,2]]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 2 - 1
[[4,3,2,2],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[5,4,3,2],[4,3,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,3,4,5,7,8],[2,6,9,10,11,12]]
 => ? = 2 - 1
[[3,3,3,2,1],[2,2,2,1]]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[1,2,3,4,8],[5,6,7,9,10]]
 => ? = 2 - 1
[[3,3,2,2,1],[2,2,1,1]]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => ? = 2 - 1
[[4,4,3,2,1],[3,3,2,1]]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[1,2,3,5,6,10],[4,7,8,9,11,12]]
 => ? = 3 - 1
[[3,2,2,2,1],[2,1,1,1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 4 - 1
[[4,3,3,2,1],[3,2,2,1]]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[1,3,4,5,6,10],[2,7,8,9,11,12]]
 => ? = 3 - 1
[[4,3,2,2,1],[3,2,1,1]]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [[1,3,4,5,8,10],[2,6,7,9,11,12]]
 => ? = 3 - 1
[[4,3,2,1,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[5,4,3,2,1],[4,3,2,1]]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [[1,3,4,5,7,8,12],[2,6,9,10,11,13,14]]
 => ? = 4 - 1
[[6,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[6,4,1],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 4 - 1
[[6,4,2],[4,2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => ? = 3 - 1
[[5,4,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[5,3,2,1],[3,1,1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => ? = 4 - 1
[[6,4,2,1],[4,2,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [[1,3,5,6,7,10],[2,4,8,9,11,12]]
 => ? = 4 - 1
[[6,4,3],[4,3]]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => ? = 2 - 1
[[5,4,3,1],[3,3,1]]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[1,2,3,5,8],[4,6,7,9,10]]
 => ? = 3 - 1
[[5,3,3,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[6,4,3,1],[4,3,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [[1,3,4,5,7,10],[2,6,8,9,11,12]]
 => ? = 4 - 1
[[5,4,3,2],[3,3,2]]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[1,2,3,5,6],[4,7,8,9,10]]
 => ? = 2 - 1
[[5,3,3,2],[3,2,2]]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => ? = 2 - 1
[[5,3,2,2],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[6,4,3,2],[4,3,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[1,3,4,5,7,8],[2,6,9,10,11,12]]
 => ? = 2 - 1
[[4,3,3,2,1],[2,2,2,1]]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[1,2,3,4,8],[5,6,7,9,10]]
 => ? = 2 - 1
[[4,3,2,2,1],[2,2,1,1]]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[1,2,3,6,8],[4,5,7,9,10]]
 => ? = 2 - 1
[[5,4,3,2,1],[3,3,2,1]]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [[1,2,3,5,6,10],[4,7,8,9,11,12]]
 => ? = 3 - 1
[[4,2,2,2,1],[2,1,1,1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => ? = 4 - 1
[[5,3,3,2,1],[3,2,2,1]]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [[1,3,4,5,6,10],[2,7,8,9,11,12]]
 => ? = 3 - 1
[[5,3,2,2,1],[3,2,1,1]]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [[1,3,4,5,8,10],[2,6,7,9,11,12]]
 => ? = 3 - 1
[[5,3,2,1,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => ? = 3 - 1
[[6,4,3,2,1],[4,3,2,1]]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [[1,3,4,5,7,8,12],[2,6,9,10,11,13,14]]
 => ? = 4 - 1
[[6,5],[5]]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => ? = 5 - 1
[[5,5,1],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 4 - 1
[[6,5,1],[5,1]]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
 => ? = 5 - 1
[[5,5,2],[4,2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => ? = 3 - 1
[[5,4,2],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => ? = 4 - 1
Description
The number of natural descents of a standard Young tableau. A natural descent of a standard tableau $T$ is an entry $i$ such that $i+1$ appears in a higher row than $i$ in English notation.
Matching statistic: St001200
Mapping
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 43%
Values
[[3,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[3,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[3,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[4,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[3,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[3,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[3,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[2,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[3,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[[3,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[2,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[2,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[3,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[3,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[3,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[[5,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[5,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,3],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[5,3],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[3,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[4,3,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,2,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[5,3,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[3,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[4,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[[4,2,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[5,3,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[3,2,2,1],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[3,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[4,3,2,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[4,2,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[4,2,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[5,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[[4,4],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[5,4],[4]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[[3,3,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,4,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[4,3,1],[3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[[5,4,1],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 4
[[3,3,2],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,4,2],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[3,2,2],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[5,4,2],[4,2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[[3,3,2,1],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[3,3,1,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,4,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[[3,2,1,1],[2]]
 => [2]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[[4,3,2,1],[3,1,1]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 4
[[4,3,1,1],[3,1]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3
[[5,4,2,1],[4,2,1]]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 4
[[3,3,3],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[[4,4,3],[3,3]]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[[4,3,3],[3,2]]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[5,4,3],[4,3]]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => ? = 2
[[2,2,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[3,3,3,1],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[3,3,2,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[[4,4,3,1],[3,3,1]]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 3
[[3,2,2,1],[2,1]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[4,3,3,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[[4,3,2,1],[3,2]]
 => [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],[4,3,1]]
 => [4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => ? = 4
[[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[3,3,3,2],[2,2,2]]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[[3,3,2,2],[2,2,1]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[[4,4,3,2],[3,3,2]]
 => [3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
[[3,2,2,2],[2,1,1]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3
[[4,3,3,2],[3,2,2]]
 => [3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 2
[[4,3,2,2],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[[5,4,3,2],[4,3,2]]
 => [4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => ? = 2
[[2,2,2,2,1],[1,1,1,1]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[[3,3,3,2,1],[2,2,2,1]]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ? = 2
[[2,2,1,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[3,3,2,2,1],[2,2,1,1]]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 2
[[4,4,3,2,1],[3,3,2,1]]
 => [3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 3
[[3,2,2,2,1],[2,1,1,1]]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 4
[[4,3,3,2,1],[3,2,2,1]]
 => [3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
 => ? = 3
[[4,3,2,2,1],[3,2,1,1]]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => ? = 3
[[4,3,2,1,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[[5,4,3,2,1],[4,3,2,1]]
 => [4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0]
 => ? = 4
[[5,2,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[4,3,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[4,2,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[5,3,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[[4,2,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[6,3,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
[[6,4,1],[4,1]]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 4
[[3,3,2],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[4,4,2],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[[6,4,2],[4,2]]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => ? = 3
[[3,3,1,1],[1,1]]
 => [1,1]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1
[[5,4,2,1],[3,2,1]]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => ? = 3
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.