Processing math: 44%

Your data matches 808 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000618
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [1]
 => 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [1]
 => 1
Description
The number of self-evacuating tableaux of given shape. This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St001196
(load all 3 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001196: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[[1,1,1,1,2,2,2],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,2,2],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,1,2,2,3],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,2,3],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,3,3],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,2,3],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,3,3],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,1,2,2,2],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,2,2],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,1,2,2,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,2,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,3],[3,3,4],[4]]
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,2,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,3],[3,3,4],[4]]
 => [6,3,1]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[[1,1,1,1,2,2,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,2,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,4,4],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,4,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,2,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,4,4],[3,3],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,4,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,1,2,3,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,3,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,3,3,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,3,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,3,3,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,1,2,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,2,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,3,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,1,2,3,4,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,2,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,3,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,2,3,4,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,3,3,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,1,2,3,3,4,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,2,2,3,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,2,3,3,3],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,2,2,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,2,3,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,2,3,4,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,3,3,3,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[[1,2,2,3,3,4,4],[3,4],[4]]
 => [7,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
Description
The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St001487
(load all 13 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 18 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0
Description
The number of connected components of a skew partition.
Matching statistic: St001107
(load all 11 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0 = 1 - 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 0 = 1 - 1
[[1],[2],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[3],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[4],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1],[5],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[3],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[4],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[2],[5],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[3],[4],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[3],[5],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[4],[5],[6]]
 => [1,1,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0 = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 0 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St001435
(load all 13 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
Description
The number of missing boxes in the first row.
Matching statistic: St001438
(load all 13 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 94%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 0 = 1 - 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [2,1]
 => [[2,1],[]]
 => 0 = 1 - 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [1,1]
 => [[1,1],[]]
 => 0 = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [3,2,1]
 => [[3,2,1],[]]
 => ? = 0 - 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St001256
(load all 49 compositions to match this statistic)
Mapping
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 92%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
[[1,1,1],[2],[3]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[1,1,2],[2],[3]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[1,1,3],[2],[3]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[1,2,2],[2],[3]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[1,2,3],[2],[3]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[1,3,3],[2],[3]]
 => [3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[[1,1],[2,2],[3]]
 => [2,2,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,1],[2,3],[3]]
 => [2,2,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1,2],[2,3],[3]]
 => [2,2,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[[1],[2],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1],[3],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1],[4],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1],[5],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2],[3],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2],[4],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[2],[5],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[3],[4],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[3],[5],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[4],[5],[6]]
 => [1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
[[1,1],[2],[5]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,1],[3],[5]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,1],[4],[5]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,2],[2],[5]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,2],[3],[5]]
 => [2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 1
[[1,1],[2,2],[3,3]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1],[2,2],[3,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1],[2,2],[4,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1],[2,3],[3,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1],[2,3],[4,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1],[3,3],[4,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,2],[2,3],[3,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,2],[2,3],[4,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,2],[3,3],[4,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[2,2],[3,3],[4,4]]
 => [2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,1],[2,2],[3,3]]
 => [3,2,2]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,2],[2,2],[3,3]]
 => [3,2,2]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,3],[2,2],[3,3]]
 => [3,2,2]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [4,3,2,1]
 => [5,5]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 0
Description
Number of simple reflexive modules that are 2-stable reflexive. See Definition 3.1. in the reference for the definition of 2-stable reflexive.
Matching statistic: St000920
(load all 6 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1],[2],[4]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1],[3],[4]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[2],[3],[4]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,3],[2],[3]]
 => [3,2,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[5]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1],[3],[5]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1],[4],[5]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[2],[3],[5]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[2],[4],[5]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[3],[4],[5]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1],[3],[4]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[1,4],[2],[4]]
 => [3,2,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[1,3],[3],[4]]
 => [4,2,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,4],[3],[4]]
 => [3,2,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[2,2],[3],[4]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[2,3],[3],[4]]
 => [4,2,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[2,4],[3],[4]]
 => [3,2,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 2 = 1 + 1
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[[1],[2],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1],[3],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1],[4],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1],[5],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[2],[3],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[2],[4],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[2],[5],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[3],[4],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[3],[5],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[4],[5],[6]]
 => [3,2,1] => [3,1,2] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1],[2],[5]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1],[3],[5]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1],[4],[5]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[[1,2],[3],[5]]
 => [4,3,1,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [10,8,9,5,6,7,1,2,3,4] => [2,3,4,10,6,7,1,9,5,8] => [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [10,8,9,4,5,6,1,2,3,7] => [2,3,7,5,6,1,10,9,4,8] => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [10,7,8,5,6,9,1,2,3,4] => [2,3,4,10,6,9,8,5,1,7] => [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [10,7,8,4,5,9,1,2,3,6] => [2,3,6,5,9,10,8,4,1,7] => [1,1,0,1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [10,6,7,4,5,8,1,2,3,9] => [2,3,9,5,8,7,4,1,10,6] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => ? = 0 + 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [10,7,8,3,4,9,1,2,5,6] => [2,5,4,9,6,10,8,3,1,7] => [1,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [10,6,7,3,4,8,1,2,5,9] => [2,5,4,8,9,7,3,1,10,6] => [1,1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 0 + 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [9,8,10,5,6,7,1,2,3,4] => [2,3,4,10,6,7,1,9,8,5] => [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [9,8,10,4,5,6,1,2,3,7] => [2,3,7,5,6,1,10,9,8,4] => [1,1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [9,7,10,5,6,8,1,2,3,4] => [2,3,4,10,6,8,9,1,7,5] => [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [9,7,10,4,5,8,1,2,3,6] => [2,3,6,5,8,10,9,1,7,4] => [1,1,0,1,0,1,1,1,0,0,1,1,0,1,1,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [9,6,10,4,5,7,1,2,3,8] => [2,3,8,5,7,9,1,10,6,4] => [1,1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,1,0,0,0]
 => ? = 0 + 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [9,7,10,3,4,8,1,2,5,6] => [2,5,4,8,6,10,9,1,7,3] => [1,1,0,1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [9,6,10,3,4,7,1,2,5,8] => [2,5,4,7,8,9,1,10,6,3] => [1,1,0,1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0,0]
 => ? = 0 + 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [8,7,9,5,6,10,1,2,3,4] => [2,3,4,10,6,9,8,7,5,1] => [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [8,7,9,4,5,10,1,2,3,6] => [2,3,6,5,9,10,8,7,4,1] => [1,1,0,1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [8,6,9,4,5,10,1,2,3,7] => [2,3,7,5,9,8,10,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [7,6,8,4,5,9,1,2,3,10] => [2,3,9,5,8,7,6,4,1,10] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [8,7,9,3,4,10,1,2,5,6] => [2,5,4,9,6,10,8,7,3,1] => [1,1,0,1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [8,6,9,3,4,10,1,2,5,7] => [2,5,4,9,7,8,10,6,3,1] => [1,1,0,1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0,0]
 => ? = 0 + 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [7,6,8,3,4,9,1,2,5,10] => [2,5,4,8,9,7,6,3,1,10] => [1,1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0,1,0]
 => ? = 0 + 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [9,6,10,5,7,8,1,2,3,4] => [2,3,4,10,7,9,8,1,6,5] => [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [9,6,10,4,7,8,1,2,3,5] => [2,3,5,7,10,9,8,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [9,5,10,4,6,7,1,2,3,8] => [2,3,8,6,9,7,1,10,5,4] => [1,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
 => ? = 0 + 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [9,6,10,3,7,8,1,2,4,5] => [2,4,7,5,10,9,8,1,6,3] => [1,1,0,1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [9,5,10,3,6,7,1,2,4,8] => [2,4,6,8,9,7,1,10,5,3] => [1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,1,0,0,0]
 => ? = 0 + 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [8,6,9,5,7,10,1,2,3,4] => [2,3,4,10,7,8,9,6,5,1] => [1,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [8,6,9,4,7,10,1,2,3,5] => [2,3,5,7,10,8,9,6,4,1] => [1,1,0,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [8,5,9,4,6,10,1,2,3,7] => [2,3,7,6,8,9,10,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [7,5,8,4,6,9,1,2,3,10] => [2,3,9,6,7,8,5,4,1,10] => [1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [8,6,9,3,7,10,1,2,4,5] => [2,4,7,5,10,8,9,6,3,1] => [1,1,0,1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [8,5,9,3,6,10,1,2,4,7] => [2,4,6,7,8,9,10,5,3,1] => [1,1,0,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [7,5,8,3,6,9,1,2,4,10] => [2,4,6,9,7,8,5,3,1,10] => [1,1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,3,5,10,1,2,6,7] => [2,6,5,8,9,7,10,4,3,1] => [1,1,0,1,1,1,1,0,0,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 0 + 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,3,5,9,1,2,6,10] => [2,6,5,7,8,9,4,3,1,10] => [1,1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 0 + 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [9,6,10,2,7,8,1,3,4,5] => [3,7,4,5,10,9,8,1,6,2] => [1,1,1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [9,5,10,2,6,7,1,3,4,8] => [3,6,4,8,9,7,1,10,5,2] => [1,1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0,0,0]
 => ? = 0 + 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [8,6,9,2,7,10,1,3,4,5] => [3,7,4,5,10,8,9,6,2,1] => [1,1,1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
 => ? = 0 + 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [8,5,9,2,6,10,1,3,4,7] => [3,6,4,7,8,9,10,5,2,1] => [1,1,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0]
 => ? = 0 + 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [7,5,8,2,6,9,1,3,4,10] => [3,6,4,9,7,8,5,2,1,10] => [1,1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 0 + 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,2,5,10,1,3,6,7] => [3,5,6,8,9,7,10,4,2,1] => [1,1,1,0,1,1,0,1,0,1,1,0,1,0,0,1,0,0,0,0]
 => ? = 0 + 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,2,5,9,1,3,6,10] => [3,5,6,7,8,9,4,2,1,10] => [1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => ? = 0 + 1
[[1,1,1,1,2,2,3],[3,3],[4]]
 => [10,7,8,1,2,3,4,5,6,9] => [2,3,4,5,6,9,8,1,10,7] => ?
 => ? = 1 + 1
[[1,1,1,2,2,2,3],[3,3],[4]]
 => [10,7,8,1,2,3,4,5,6,9] => [2,3,4,5,6,9,8,1,10,7] => ?
 => ? = 1 + 1
[[1,1,1,2,2,3,3],[3,3],[4]]
 => [10,6,7,1,2,3,4,5,8,9] => [2,3,4,5,8,7,1,9,10,6] => ?
 => ? = 1 + 1
[[1,1,2,2,2,2,3],[3,3],[4]]
 => [10,7,8,1,2,3,4,5,6,9] => [2,3,4,5,6,9,8,1,10,7] => ?
 => ? = 1 + 1
[[1,1,2,2,2,3,3],[3,3],[4]]
 => [10,6,7,1,2,3,4,5,8,9] => [2,3,4,5,8,7,1,9,10,6] => ?
 => ? = 1 + 1
[[1,1,1,1,2,2,3],[3,4],[4]]
 => [9,7,10,1,2,3,4,5,6,8] => [2,3,4,5,6,8,9,10,7,1] => ?
 => ? = 1 + 1
[[1,1,1,2,2,2,3],[3,4],[4]]
 => [9,7,10,1,2,3,4,5,6,8] => [2,3,4,5,6,8,9,10,7,1] => ?
 => ? = 1 + 1
[[1,1,1,2,2,3],[3,3,4],[4]]
 => [9,6,7,10,1,2,3,4,5,8] => [2,3,4,5,8,7,9,10,6,1] => ?
 => ? = 1 + 1
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
Matching statistic: St000629
(load all 22 compositions to match this statistic)
Mapping
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000629: Binary words ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Values
[[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1],[2],[4]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1],[3],[4]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[2],[3],[4]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [3,1,4,2] => 011 => 0 = 1 - 1
[[1,3],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 110 => 0 = 1 - 1
[[1],[2],[5]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1],[3],[5]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1],[4],[5]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[2],[3],[5]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[2],[4],[5]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[3],[4],[5]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,1],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => 011 => 0 = 1 - 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [3,1,4,2] => 011 => 0 = 1 - 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 110 => 0 = 1 - 1
[[1,4],[2],[4]]
 => [3,2,1,4] => [3,2,1,4] => 110 => 0 = 1 - 1
[[1,3],[3],[4]]
 => [4,2,1,3] => [3,1,4,2] => 011 => 0 = 1 - 1
[[1,4],[3],[4]]
 => [3,2,1,4] => [3,2,1,4] => 110 => 0 = 1 - 1
[[2,2],[3],[4]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[2,3],[3],[4]]
 => [4,2,1,3] => [3,1,4,2] => 011 => 0 = 1 - 1
[[2,4],[3],[4]]
 => [3,2,1,4] => [3,2,1,4] => 110 => 0 = 1 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 111 => 0 = 1 - 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [1,2,5,4,3] => 0011 => 0 = 1 - 1
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [1,4,2,5,3] => 0011 => 0 = 1 - 1
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [1,4,3,2,5] => 0110 => 0 = 1 - 1
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [3,1,2,5,4] => 0101 => 0 = 1 - 1
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [3,1,4,2,5] => 0110 => 0 = 1 - 1
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => 1100 => 0 = 1 - 1
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [1,5,2,4,3] => 0011 => 0 = 1 - 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [1,5,3,2,4] => 0101 => 0 = 1 - 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [1,4,5,2,3] => 0001 => 0 = 1 - 1
[[1],[2],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1],[3],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1],[4],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1],[5],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[2],[3],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[2],[4],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[2],[5],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[3],[4],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[3],[5],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[4],[5],[6]]
 => [3,2,1] => [3,2,1] => 11 => 0 = 1 - 1
[[1,1],[2],[5]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,1],[3],[5]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,1],[4],[5]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,2],[2],[5]]
 => [4,2,1,3] => [3,1,4,2] => 011 => 0 = 1 - 1
[[1,2],[3],[5]]
 => [4,3,1,2] => [1,4,3,2] => 011 => 0 = 1 - 1
[[1,1,1,1],[2,2,2],[3,3],[4]]
 => [10,8,9,5,6,7,1,2,3,4] => [1,2,3,10,4,5,9,6,8,7] => ? => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,3],[4]]
 => [10,8,9,4,5,6,1,2,3,7] => [1,2,9,3,4,5,10,6,8,7] => ? => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,3],[4]]
 => [10,7,8,5,6,9,1,2,3,4] => [1,2,3,10,4,8,5,6,9,7] => ? => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,3],[4]]
 => [10,7,8,4,5,9,1,2,3,6] => [1,2,3,5,9,10,4,6,8,7] => ? => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,3],[4]]
 => [10,6,7,4,5,8,1,2,3,9] => [1,2,9,3,7,4,5,6,10,8] => ? => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,3],[4]]
 => [10,7,8,3,4,9,1,2,5,6] => [1,2,4,8,3,10,5,6,9,7] => ? => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,3],[4]]
 => [10,6,7,3,4,8,1,2,5,9] => [1,2,4,8,9,3,5,6,10,7] => ? => ? = 0 - 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
 => [9,8,10,5,6,7,1,2,3,4] => [1,2,3,10,4,5,9,7,6,8] => ? => ? = 0 - 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
 => [9,8,10,4,5,6,1,2,3,7] => [1,2,9,3,4,5,10,7,6,8] => ? => ? = 0 - 1
[[1,1,1,1],[2,2,3],[3,4],[4]]
 => [9,7,10,5,6,8,1,2,3,4] => [1,2,3,10,4,5,8,9,6,7] => ? => ? = 0 - 1
[[1,1,1,2],[2,2,3],[3,4],[4]]
 => [9,7,10,4,5,8,1,2,3,6] => [1,2,3,5,6,10,8,9,4,7] => ? => ? = 0 - 1
[[1,1,1,3],[2,2,3],[3,4],[4]]
 => [9,6,10,4,5,7,1,2,3,8] => [1,2,9,3,4,7,5,10,6,8] => ? => ? = 0 - 1
[[1,1,2,2],[2,2,3],[3,4],[4]]
 => [9,7,10,3,4,8,1,2,5,6] => [1,2,4,5,3,10,8,9,6,7] => ? => ? = 0 - 1
[[1,1,2,3],[2,2,3],[3,4],[4]]
 => [9,6,10,3,4,7,1,2,5,8] => [1,2,4,5,9,7,3,10,6,8] => ? => ? = 0 - 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
 => [8,7,9,5,6,10,1,2,3,4] => [1,2,3,10,4,8,6,5,7,9] => ? => ? = 0 - 1
[[1,1,1,2],[2,2,4],[3,4],[4]]
 => [8,7,9,4,5,10,1,2,3,6] => [1,2,3,5,9,10,6,4,7,8] => ? => ? = 0 - 1
[[1,1,1,3],[2,2,4],[3,4],[4]]
 => [8,6,9,4,5,10,1,2,3,7] => [1,2,3,5,9,7,10,4,6,8] => ? => ? = 0 - 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
 => [7,6,8,4,5,9,1,2,3,10] => [1,2,9,3,7,5,4,6,8,10] => ? => ? = 0 - 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
 => [8,7,9,3,4,10,1,2,5,6] => [1,2,4,8,3,10,6,5,7,9] => ? => ? = 0 - 1
[[1,1,2,3],[2,2,4],[3,4],[4]]
 => [8,6,9,3,4,10,1,2,5,7] => [1,2,4,8,3,7,10,5,6,9] => ? => ? = 0 - 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
 => [7,6,8,3,4,9,1,2,5,10] => [1,2,4,8,9,5,3,6,7,10] => ? => ? = 0 - 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
 => [9,6,10,5,7,8,1,2,3,4] => [1,2,3,10,4,7,5,9,6,8] => ? => ? = 0 - 1
[[1,1,1,2],[2,3,3],[3,4],[4]]
 => [9,6,10,4,7,8,1,2,3,5] => [1,2,3,5,10,7,4,9,6,8] => ? => ? = 0 - 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
 => [9,5,10,4,6,7,1,2,3,8] => [1,2,9,3,6,4,5,10,7,8] => ? => ? = 0 - 1
[[1,1,2,2],[2,3,3],[3,4],[4]]
 => [9,6,10,3,7,8,1,2,4,5] => [1,2,4,3,10,7,5,9,6,8] => ? => ? = 0 - 1
[[1,1,2,3],[2,3,3],[3,4],[4]]
 => [9,5,10,3,6,7,1,2,4,8] => [1,2,4,9,6,3,5,10,7,8] => ? => ? = 0 - 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
 => [8,6,9,5,7,10,1,2,3,4] => [1,2,3,10,4,7,8,5,6,9] => ? => ? = 0 - 1
[[1,1,1,2],[2,3,4],[3,4],[4]]
 => [8,6,9,4,7,10,1,2,3,5] => [1,2,3,5,10,7,8,4,6,9] => ? => ? = 0 - 1
[[1,1,1,3],[2,3,4],[3,4],[4]]
 => [8,5,9,4,6,10,1,2,3,7] => [1,2,3,5,8,9,10,4,6,7] => ? => ? = 0 - 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
 => [7,5,8,4,6,9,1,2,3,10] => [1,2,9,3,6,7,4,5,8,10] => ? => ? = 0 - 1
[[1,1,2,2],[2,3,4],[3,4],[4]]
 => [8,6,9,3,7,10,1,2,4,5] => [1,2,4,3,10,7,8,5,6,9] => ? => ? = 0 - 1
[[1,1,2,3],[2,3,4],[3,4],[4]]
 => [8,5,9,3,6,10,1,2,4,7] => [1,2,4,3,8,9,10,5,6,7] => ? => ? = 0 - 1
[[1,1,2,4],[2,3,4],[3,4],[4]]
 => [7,5,8,3,6,9,1,2,4,10] => [1,2,4,9,6,7,3,5,8,10] => ? => ? = 0 - 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,3,5,10,1,2,6,7] => [1,2,4,7,8,3,10,5,6,9] => ? => ? = 0 - 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,3,5,9,1,2,6,10] => [1,2,4,7,8,9,3,5,6,10] => ? => ? = 0 - 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
 => [9,6,10,2,7,8,1,3,4,5] => [1,3,2,4,10,7,5,9,6,8] => ? => ? = 0 - 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
 => [9,5,10,2,6,7,1,3,4,8] => [1,3,2,9,6,4,5,10,7,8] => ? => ? = 0 - 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
 => [8,6,9,2,7,10,1,3,4,5] => [1,3,2,4,10,7,8,5,6,9] => ? => ? = 0 - 1
[[1,2,2,3],[2,3,4],[3,4],[4]]
 => [8,5,9,2,6,10,1,3,4,7] => [1,3,2,4,8,9,10,5,6,7] => ? => ? = 0 - 1
[[1,2,2,4],[2,3,4],[3,4],[4]]
 => [7,5,8,2,6,9,1,3,4,10] => [1,3,2,9,6,7,4,5,8,10] => ? => ? = 0 - 1
[[1,2,3,3],[2,3,4],[3,4],[4]]
 => [8,4,9,2,5,10,1,3,6,7] => [1,3,2,7,8,4,10,5,6,9] => ? => ? = 0 - 1
[[1,2,3,4],[2,3,4],[3,4],[4]]
 => [7,4,8,2,5,9,1,3,6,10] => [1,3,2,7,8,9,4,5,6,10] => ? => ? = 0 - 1
[[1,1,1,1,2,2,2],[3,3],[4]]
 => [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,7,9,8] => ? => ? = 1 - 1
[[1,1,1,2,2,2,2],[3,3],[4]]
 => [10,8,9,1,2,3,4,5,6,7] => [1,2,3,4,5,6,10,7,9,8] => ? => ? = 1 - 1
[[1,1,1,2,2,3,3],[3,3],[4]]
 => [10,6,7,1,2,3,4,5,8,9] => ? => ? => ? = 1 - 1
[[1,1,2,2,2,3,3],[3,3],[4]]
 => [10,6,7,1,2,3,4,5,8,9] => ? => ? => ? = 1 - 1
[[1,1,1,1,2,2,2],[3,4],[4]]
 => [9,8,10,1,2,3,4,5,6,7] => ? => ? => ? = 1 - 1
[[1,1,1,2,2,2,2],[3,4],[4]]
 => [9,8,10,1,2,3,4,5,6,7] => ? => ? => ? = 1 - 1
[[1,1,1,1,2,2,3],[3,4],[4]]
 => [9,7,10,1,2,3,4,5,6,8] => ? => ? => ? = 1 - 1
[[1,1,1,2,2,2,3],[3,4],[4]]
 => [9,7,10,1,2,3,4,5,6,8] => ? => ? => ? = 1 - 1
Description
The defect of a binary word. The defect of a finite word $w$ is given by the difference between the maximum possible number and the actual number of palindromic factors contained in $w$. The maximum possible number of palindromic factors in a word $w$ is $|w|+1$.
The following 798 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000929The constant term of the character polynomial of an integer partition. St000659The number of rises of length at least 2 of a Dyck path. St000097The order of the largest clique of the graph. St000352The Elizalde-Pak rank of a permutation. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000546The number of global descents of a permutation. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000775The multiplicity of the largest eigenvalue in a graph. St000805The number of peaks of the associated bargraph. St000919The number of maximal left branches of a binary tree. St000990The first ascent of a permutation. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001468The smallest fixpoint of a permutation. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001722The number of minimal chains with small intervals between a binary word and the top element. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000210Minimum over maximum difference of elements in cycles. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000516The number of stretching pairs of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000623The number of occurrences of the pattern 52341 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001793The difference between the clique number and the chromatic number of a graph. St001847The number of occurrences of the pattern 1432 in a permutation. St001957The number of Hasse diagrams with a given underlying undirected graph. St000264The girth of a graph, which is not a tree. St000011The number of touch points (or returns) of a Dyck path. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000054The first entry of the permutation. St000069The number of maximal elements of a poset. St000253The crossing number of a set partition. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000382The first part of an integer composition. St000456The monochromatic index of a connected graph. St000504The cardinality of the first block of a set partition. St000505The biggest entry in the block containing the 1. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000570The Edelman-Greene number of a permutation. St000627The exponent of a binary word. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000660The number of rises of length at least 3 of a Dyck path. St000667The greatest common divisor of the parts of the partition. St000679The pruning number of an ordered tree. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000701The protection number of a binary tree. St000710The number of big deficiencies of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000729The minimal arc length of a set partition. St000735The last entry on the main diagonal of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000758The length of the longest staircase fitting into an integer composition. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000781The number of proper colouring schemes of a Ferrers diagram. St000785The number of distinct colouring schemes of a graph. St000816The number of standard composition tableaux of the composition. St000823The number of unsplittable factors of the set partition. St000845The maximal number of elements covered by an element in a poset. St000883The number of longest increasing subsequences of a permutation. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000913The number of ways to refine the partition into singletons. St000948The chromatic discriminant of a graph. St000971The smallest closer of a set partition. St000993The multiplicity of the largest part of an integer partition. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001075The minimal size of a block of a set partition. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001271The competition number of a graph. St001272The number of graphs with the same degree sequence. St001277The degeneracy of a graph. St001316The domatic number of a graph. St001344The neighbouring number of a permutation. St001358The largest degree of a regular subgraph of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001363The Euler characteristic of a graph according to Knill. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001393The induced matching number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001518The number of graphs with the same ordinary spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001597The Frobenius rank of a skew partition. St001732The number of peaks visible from the left. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000002The number of occurrences of the pattern 123 in a permutation. St000017The number of inversions of a standard tableau. St000052The number of valleys of a Dyck path not on the x-axis. St000058The order of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000091The descent variation of a composition. St000098The chromatic number of a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000124The cardinality of the preimage of the Simion-Schmidt map. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000268The number of strongly connected orientations of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000296The length of the symmetric border of a binary word. St000298The order dimension or Dushnik-Miller dimension of a poset. St000312The number of leaves in a graph. St000317The cycle descent number of a permutation. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000351The determinant of the adjacency matrix of a graph. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000379The number of Hamiltonian cycles in a graph. St000386The number of factors DDU in a Dyck path. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000403The Szeged index minus the Wiener index of a graph. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000439The position of the first down step of a Dyck path. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000637The length of the longest cycle in a graph. St000658The number of rises of length 2 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000674The number of hills of a Dyck path. St000699The toughness times the least common multiple of 1,. St000709The number of occurrences of 14-2-3 or 14-3-2. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000761The number of ascents in an integer composition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000842The breadth of a permutation. St000862The number of parts of the shifted shape of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000974The length of the trunk of an ordered tree. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000989The number of final rises of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001062The maximal size of a block of a set partition. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001119The length of a shortest maximal path in a graph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001261The Castelnuovo-Mumford regularity of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001330The hat guessing number of a graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001353The number of prime nodes in the modular decomposition of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001356The number of vertices in prime modules of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001381The fertility of a permutation. St001394The genus of a permutation. St001403The number of vertical separators in a permutation. St001434The number of negative sum pairs of a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001471The magnitude of a Dyck path. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001494The Alon-Tarsi number of a graph. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001580The acyclic chromatic number of a graph. St001584The area statistic between a Dyck path and its bounce path. St001593This is the number of standard Young tableaux of the given shifted shape. St001596The number of two-by-two squares inside a skew partition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001716The 1-improper chromatic number of a graph. St001730The number of times the path corresponding to a binary word crosses the base line. St001736The total number of cycles in a graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001108The 2-dynamic chromatic number of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000552The number of cut vertices of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001826The maximal number of leaves on a vertex of a graph. St000441The number of successions of a permutation. St000534The number of 2-rises of a permutation. St000665The number of rafts of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St000886The number of permutations with the same antidiagonal sums. St000358The number of occurrences of the pattern 31-2. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000647The number of big descents of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000963The 2-shifted major index of a permutation. St000909The number of maximal chains of maximal size in a poset. St000068The number of minimal elements in a poset. St001307The number of induced stars on four vertices in a graph. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000834The number of right outer peaks of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000007The number of saliances of the permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000042The number of crossings of a perfect matching. St000214The number of adjacencies of a permutation. St000217The number of occurrences of the pattern 312 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000252The number of nodes of degree 3 of a binary tree. St000356The number of occurrences of the pattern 13-2. St000649The number of 3-excedences of a permutation. St000733The row containing the largest entry of a standard tableau. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001083The number of boxed occurrences of 132 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001411The number of patterns 321 or 3412 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001479The number of bridges of a graph. St000788The number of nesting-similar perfect matchings of a perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000640The rank of the largest boolean interval in a poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001260The permanent of an alternating sign matrix. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000463The number of admissible inversions of a permutation. St000871The number of very big ascents of a permutation. St000961The shifted major index of a permutation. St001301The first Betti number of the order complex associated with the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001613The binary logarithm of the size of the center of a lattice. St001621The number of atoms of a lattice. St001665The number of pure excedances of a permutation. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001846The number of elements which do not have a complement in the lattice. St001850The number of Hecke atoms of a permutation. St000663The number of right floats of a permutation. St001947The number of ties in a parking function. St001718The number of non-empty open intervals in a poset. St000654The first descent of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000093The cardinality of a maximal independent set of vertices of a graph. St000096The number of spanning trees of a graph. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000314The number of left-to-right-maxima of a permutation. St000338The number of pixed points of a permutation. St000349The number of different adjacency matrices of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000450The number of edges minus the number of vertices plus 2 of a graph. St000472The sum of the ascent bottoms of a permutation. St000553The number of blocks of a graph. St000646The number of big ascents of a permutation. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000873The aix statistic of a permutation. St000916The packing number of a graph. St000917The open packing number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000991The number of right-to-left minima of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001282The number of graphs with the same chromatic polynomial. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001512The minimum rank of a graph. St001642The Prague dimension of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001672The restrained domination number of a graph. St001734The lettericity of a graph. St001735The number of permutations with the same set of runs. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001765The number of connected components of the friends and strangers graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001828The Euler characteristic of a graph. St001829The common independence number of a graph. St001890The maximum magnitude of the Möbius function of a poset. St001917The order of toric promotion on the set of labellings of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000039The number of crossings of a permutation. St000051The size of the left subtree of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000095The number of triangles of a graph. St000133The "bounce" of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000258The burning number of a graph. St000274The number of perfect matchings of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000308The height of the tree associated to a permutation. St000315The number of isolated vertices of a graph. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000461The rix statistic of a permutation. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000624The normalized sum of the minimal distances to a greater element. St000650The number of 3-rises of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000822The Hadwiger number of the graph. St000918The 2-limited packing number of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001060The distinguishing index of a graph. St001093The detour number of a graph. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001308The number of induced paths on three vertices in a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001323The independence gap of a graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001374The Padmakar-Ivan index of a graph. St001429The number of negative entries in a signed permutation. St001519The pinnacle sum of a permutation. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001674The number of vertices of the largest induced star graph in the graph. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001708The number of pairs of vertices of different degree in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001764The number of non-convex subsets of vertices in a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001799The number of proper separations of a graph. St001871The number of triconnected components of a graph. St000451The length of the longest pattern of the form k 1 2. St000455The second largest eigenvalue of a graph if it is integral. St001703The villainy of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000022The number of fixed points of a permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000694The number of affine bounded permutations that project to a given permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000508Eigenvalues of the random-to-random operator acting on a simple module. St001557The number of inversions of the second entry of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001948The number of augmented double ascents of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001081The number of minimal length factorizations of a permutation into star transpositions. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001430The number of positive entries in a signed permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001488The number of corners of a skew partition. St000116The major index of a semistandard tableau obtained by standardizing. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001868The number of alignments of type NE of a signed permutation. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001555The order of a signed permutation. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000850The number of 1/2-balanced pairs in a poset. St000911The number of maximal antichains of maximal size in a poset. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St001770The number of facets of a certain subword complex associated with the signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function. St000633The size of the automorphism group of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001399The distinguishing number of a poset. St001423The number of distinct cubes in a binary word. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001851The number of Hecke atoms of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000717The number of ordinal summands of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001892The flag excedance statistic of a signed permutation. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001964The interval resolution global dimension of a poset. St001618The cardinality of the Frattini sublattice of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000636The hull number of a graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001638The book thickness of a graph. St001654The monophonic hull number of a graph. St001410The minimal entry of a semistandard tableau. St001510The number of self-evacuating linear extensions of a finite poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001268The size of the largest ordinal summand in the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St000635The number of strictly order preserving maps of a poset into itself. St001827The number of two-component spanning forests of a graph. St000307The number of rowmotion orbits of a poset. St001625The Möbius invariant of a lattice. St000741The Colin de Verdière graph invariant. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. St000084The number of subtrees. St000100The number of linear extensions of a poset. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St001623The number of doubly irreducible elements of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001754The number of tolerances of a finite lattice. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000094The depth of an ordered tree. St000189The number of elements in the poset. St000327The number of cover relations in a poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001645The pebbling number of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001879The 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000180The number of chains of a poset. St000400The path length of an ordered tree. St001909The number of interval-closed sets of a poset. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000529The number of permutations whose descent word is the given binary word. St000416The number of inequivalent increasing trees of an ordered tree. St000634The number of endomorphisms of a poset. St000410The tree factorial of an ordered tree. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.