Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St000225
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1]
 => [1]
 => 0
[[1],[1]]
 => [1,1]
 => [2]
 => 0
[[2]]
 => [2]
 => [1,1]
 => 0
[[1,1]]
 => [2]
 => [1,1]
 => 0
[[1],[1],[1]]
 => [1,1,1]
 => [3]
 => 0
[[2],[1]]
 => [2,1]
 => [2,1]
 => 1
[[1,1],[1]]
 => [2,1]
 => [2,1]
 => 1
[[3]]
 => [3]
 => [1,1,1]
 => 0
[[2,1]]
 => [3]
 => [1,1,1]
 => 0
[[1,1,1]]
 => [3]
 => [1,1,1]
 => 0
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [4]
 => 0
[[2],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 2
[[2],[2]]
 => [2,2]
 => [2,2]
 => 0
[[1,1],[1],[1]]
 => [2,1,1]
 => [3,1]
 => 2
[[1,1],[1,1]]
 => [2,2]
 => [2,2]
 => 0
[[3],[1]]
 => [3,1]
 => [2,1,1]
 => 1
[[2,1],[1]]
 => [3,1]
 => [2,1,1]
 => 1
[[1,1,1],[1]]
 => [3,1]
 => [2,1,1]
 => 1
[[4]]
 => [4]
 => [1,1,1,1]
 => 0
[[3,1]]
 => [4]
 => [1,1,1,1]
 => 0
[[2,2]]
 => [4]
 => [1,1,1,1]
 => 0
[[2,1,1]]
 => [4]
 => [1,1,1,1]
 => 0
[[1,1,1,1]]
 => [4]
 => [1,1,1,1]
 => 0
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [5]
 => 0
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 3
[[2],[2],[1]]
 => [2,2,1]
 => [3,2]
 => 1
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [4,1]
 => 3
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [3,2]
 => 1
[[3],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 2
[[3],[2]]
 => [3,2]
 => [2,2,1]
 => 1
[[2,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 2
[[2,1],[2]]
 => [3,2]
 => [2,2,1]
 => 1
[[2,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 1
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [3,1,1]
 => 2
[[1,1,1],[1,1]]
 => [3,2]
 => [2,2,1]
 => 1
[[4],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[3,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[2,2],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[2,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[1,1,1,1],[1]]
 => [4,1]
 => [2,1,1,1]
 => 1
[[5]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[4,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[3,2]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[3,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[2,2,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[2,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[1,1,1,1,1]]
 => [5]
 => [1,1,1,1,1]
 => 0
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [6]
 => 0
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [5,1]
 => 4
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [4,2]
 => 2
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St001804
Mapping
Mp00311: Plane partitions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 78%
Values
[[1]]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[[1],[1]]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[[2]]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[[1,1]]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[[1],[1],[1]]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[[2],[1]]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[[1,1],[1]]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[[3]]
 => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[[2,1]]
 => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[[1,1,1]]
 => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[[1],[1],[1],[1]]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[[2],[1],[1]]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[[2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 0 + 1
[[1,1],[1],[1]]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[[1,1],[1,1]]
 => [2,2]
 => [[1,2],[3,4]]
 => 1 = 0 + 1
[[3],[1]]
 => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[[2,1],[1]]
 => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[[1,1,1],[1]]
 => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[[4]]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[[3,1]]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[[2,2]]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[[2,1,1]]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[[1,1,1,1]]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[[1],[1],[1],[1],[1]]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[[2],[1],[1],[1]]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[[2],[2],[1]]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 1 + 1
[[1,1],[1],[1],[1]]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[[1,1],[1,1],[1]]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2 = 1 + 1
[[3],[1],[1]]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[[3],[2]]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 2 = 1 + 1
[[2,1],[1],[1]]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[[2,1],[2]]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 2 = 1 + 1
[[2,1],[1,1]]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 2 = 1 + 1
[[1,1,1],[1],[1]]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[[1,1,1],[1,1]]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 2 = 1 + 1
[[4],[1]]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[[3,1],[1]]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[[2,2],[1]]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[[2,1,1],[1]]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[[1,1,1,1],[1]]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[[5]]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[4,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[3,2]]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[3,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[2,2,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[2,1,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[1,1,1,1,1]]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1 = 0 + 1
[[2],[1],[1],[1],[1]]
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 5 = 4 + 1
[[2],[2],[1],[1]]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3 = 2 + 1
[[1],[1],[1],[1],[1],[1],[1],[1],[1]]
 => [1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
 => ? = 0 + 1
[[2],[1],[1],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1,1,1]
 => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[[2],[2],[1],[1],[1],[1],[1]]
 => [2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3],[4],[5],[6],[8]]
 => ? = 5 + 1
[[2],[2],[2],[1],[1],[1]]
 => [2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4],[6],[8]]
 => ? = 3 + 1
[[2],[2],[2],[2],[1]]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => ? = 1 + 1
[[1,1],[1],[1],[1],[1],[1],[1],[1]]
 => [2,1,1,1,1,1,1,1]
 => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
 => ? = 7 + 1
[[1,1],[1,1],[1],[1],[1],[1],[1]]
 => [2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3],[4],[5],[6],[8]]
 => ? = 5 + 1
[[1,1],[1,1],[1,1],[1],[1],[1]]
 => [2,2,2,1,1,1]
 => [[1,5],[2,7],[3,9],[4],[6],[8]]
 => ? = 3 + 1
[[1,1],[1,1],[1,1],[1,1],[1]]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => ? = 1 + 1
[[3],[1],[1],[1],[1],[1],[1]]
 => [3,1,1,1,1,1,1]
 => [[1,8,9],[2],[3],[4],[5],[6],[7]]
 => ? = 6 + 1
[[3],[2],[1],[1],[1],[1]]
 => [3,2,1,1,1,1]
 => [[1,6,9],[2,8],[3],[4],[5],[7]]
 => ? = 5 + 1
[[3],[2],[2],[1],[1]]
 => [3,2,2,1,1]
 => [[1,4,9],[2,6],[3,8],[5],[7]]
 => ? = 4 + 1
[[3],[2],[2],[2]]
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? = 3 + 1
[[3],[3],[1],[1],[1]]
 => [3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => ? = 3 + 1
[[3],[3],[2],[1]]
 => [3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => ? = 2 + 1
[[3],[3],[3]]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => ? = 0 + 1
[[2,1],[1],[1],[1],[1],[1],[1]]
 => [3,1,1,1,1,1,1]
 => [[1,8,9],[2],[3],[4],[5],[6],[7]]
 => ? = 6 + 1
[[2,1],[2],[1],[1],[1],[1]]
 => [3,2,1,1,1,1]
 => [[1,6,9],[2,8],[3],[4],[5],[7]]
 => ? = 5 + 1
[[2,1],[2],[2],[1],[1]]
 => [3,2,2,1,1]
 => [[1,4,9],[2,6],[3,8],[5],[7]]
 => ? = 4 + 1
[[2,1],[2],[2],[2]]
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? = 3 + 1
[[2,1],[1,1],[1],[1],[1],[1]]
 => [3,2,1,1,1,1]
 => [[1,6,9],[2,8],[3],[4],[5],[7]]
 => ? = 5 + 1
[[2,1],[1,1],[1,1],[1],[1]]
 => [3,2,2,1,1]
 => [[1,4,9],[2,6],[3,8],[5],[7]]
 => ? = 4 + 1
[[2,1],[1,1],[1,1],[1,1]]
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? = 3 + 1
[[2,1],[2,1],[1],[1],[1]]
 => [3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => ? = 3 + 1
[[2,1],[2,1],[2],[1]]
 => [3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => ? = 2 + 1
[[2,1],[2,1],[1,1],[1]]
 => [3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => ? = 2 + 1
[[2,1],[2,1],[2,1]]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => ? = 0 + 1
[[1,1,1],[1],[1],[1],[1],[1],[1]]
 => [3,1,1,1,1,1,1]
 => [[1,8,9],[2],[3],[4],[5],[6],[7]]
 => ? = 6 + 1
[[1,1,1],[1,1],[1],[1],[1],[1]]
 => [3,2,1,1,1,1]
 => [[1,6,9],[2,8],[3],[4],[5],[7]]
 => ? = 5 + 1
[[1,1,1],[1,1],[1,1],[1],[1]]
 => [3,2,2,1,1]
 => [[1,4,9],[2,6],[3,8],[5],[7]]
 => ? = 4 + 1
[[1,1,1],[1,1],[1,1],[1,1]]
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? = 3 + 1
[[1,1,1],[1,1,1],[1],[1],[1]]
 => [3,3,1,1,1]
 => [[1,5,6],[2,8,9],[3],[4],[7]]
 => ? = 3 + 1
[[1,1,1],[1,1,1],[1,1],[1]]
 => [3,3,2,1]
 => [[1,3,6],[2,5,9],[4,8],[7]]
 => ? = 2 + 1
[[1,1,1],[1,1,1],[1,1,1]]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => ? = 0 + 1
[[4],[1],[1],[1],[1],[1]]
 => [4,1,1,1,1,1]
 => [[1,7,8,9],[2],[3],[4],[5],[6]]
 => ? = 5 + 1
[[4],[2],[1],[1],[1]]
 => [4,2,1,1,1]
 => [[1,5,8,9],[2,7],[3],[4],[6]]
 => ? = 4 + 1
[[4],[2],[2],[1]]
 => [4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => ? = 3 + 1
[[4],[3],[1],[1]]
 => [4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => ? = 3 + 1
[[4],[3],[2]]
 => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => ? = 2 + 1
[[4],[4],[1]]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => ? = 1 + 1
[[3,1],[1],[1],[1],[1],[1]]
 => [4,1,1,1,1,1]
 => [[1,7,8,9],[2],[3],[4],[5],[6]]
 => ? = 5 + 1
[[3,1],[2],[1],[1],[1]]
 => [4,2,1,1,1]
 => [[1,5,8,9],[2,7],[3],[4],[6]]
 => ? = 4 + 1
[[3,1],[2],[2],[1]]
 => [4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => ? = 3 + 1
[[3,1],[1,1],[1],[1],[1]]
 => [4,2,1,1,1]
 => [[1,5,8,9],[2,7],[3],[4],[6]]
 => ? = 4 + 1
[[3,1],[1,1],[1,1],[1]]
 => [4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => ? = 3 + 1
[[3,1],[3],[1],[1]]
 => [4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => ? = 3 + 1
[[3,1],[3],[2]]
 => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => ? = 2 + 1
[[3,1],[2,1],[1],[1]]
 => [4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => ? = 3 + 1
[[3,1],[2,1],[2]]
 => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => ? = 2 + 1
[[3,1],[2,1],[1,1]]
 => [4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => ? = 2 + 1
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.