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Your data matches 48 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St001464

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[2,2]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 0 + 1
[[1],[2]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[1,3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[2,3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[3,3]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 0 + 1
[[1],[3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[2],[3]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[1,1,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[1,2,2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[2,2,2]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 0 + 1
[[1,1],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[1,2],[2]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[1,4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[2,4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[3,4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[4,4]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 0 + 1
[[1],[4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[2],[4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[3],[4]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[1,1,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[1,2,3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[[1,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[2,2,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[2,3,3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[3,3,3]]
 => [3]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1 = 0 + 1
[[1,1],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[1,2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[[1,3],[2]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[[1,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[2,2],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[2,3],[3]]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[[1],[2],[3]]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 3 = 2 + 1
[[1,1,1,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 1 + 1
[[1,1,2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 1 + 1
[[1,2,2,2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 1 + 1
[[2,2,2,2]]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1 = 0 + 1
[[1,1,1],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 1 + 1
[[1,1,2],[2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 1 + 1
[[1,2,2],[2]]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2 = 1 + 1
[[1,1],[2,2]]
 => [2,2]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2 = 1 + 1
[[1,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[2,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[3,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[4,5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[5,5]]
 => [2]
 => [1,0,1,0]
 => [1,2] => 1 = 0 + 1
[[1],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[2],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[3],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[4],[5]]
 => [1,1]
 => [1,1,0,0]
 => [2,1] => 2 = 1 + 1
[[1,1,1,1,1,1,1,2]]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? ∊ {0,1,1,1,1} + 1
[[1,2,2,2,2,2,2,2]]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? ∊ {0,1,1,1,1} + 1
[[2,2,2,2,2,2,2,2]]
 => [8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6,7,8] => ? ∊ {0,1,1,1,1} + 1
[[1,1,1,1,1,1,1],[2]]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? ∊ {0,1,1,1,1} + 1
[[1,2,2,2,2,2,2],[2]]
 => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? ∊ {0,1,1,1,1} + 1

Description

The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.

Matching statistic: St000063

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000063: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 99%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition

$\lambda$ also counts cover-inclusive Dyck tilings of

$\lambda\setminus\mu$ , summed over all

$\mu$ , as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

Matching statistic: St000108

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000108: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 99%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of partitions contained in the given partition.

Matching statistic: St000532

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000532: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 99%●distinct values known / distinct values provided: 80%

Values

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

Description

The total number of rook placements on a Ferrers board.

Matching statistic: St000738

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 99%●distinct values known / distinct values provided: 80%

Values

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

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St001389

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 99%●distinct values known / distinct values provided: 80%

Values

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

Description

The number of partitions of the same length below the given integer partition. For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

Matching statistic: St001400

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001400: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 99%●distinct values known / distinct values provided: 80%

Values

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

Description

The total number of Littlewood-Richardson tableaux of given shape. This is the multiplicity of the Schur function

$s_\lambda$ in

$\sum_{\mu, \nu} s_\mu s_\nu$ , where the sum is over all partitions

$\mu$ and

$\nu$ .

Matching statistic: St000864

Mapping

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000864: Permutations ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%

Values

[[1,2]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,2]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[2]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[1,3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3,3]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2],[3]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[1,1,2]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[1,2,2]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[2,2,2]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[[1,1],[2]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[1,2],[2]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[1,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3,4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[4,4]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3],[4]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[1,1,3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[1,2,3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,3,3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[2,2,3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[2,3,3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[3,3,3]]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 0
[[1,1],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[1,2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,3],[2]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,3],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[2,2],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[2,3],[3]]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[[1],[2],[3]]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 2
[[1,1,1,2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[[1,1,2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[[1,2,2,2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[[2,2,2,2]]
 => [4]
 => [[1,2,3,4]]
 => [1,2,3,4] => 0
[[1,1,1],[2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[[1,1,2],[2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[[1,2,2],[2]]
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 1
[[1,1],[2,2]]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 1
[[1,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[4,5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[5,5]]
 => [2]
 => [[1,2]]
 => [1,2] => 0
[[1],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[2],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[3],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[4],[5]]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 1
[[1,1,1,1,1,1,2]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,1,2,2]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,2,2,2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,2,2,2,2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,2,2,2,2,2]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,2,2,2,2,2]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,1,1],[2]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,1,2],[2]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,2,2],[2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,2,2,2],[2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,2,2,2,2],[2]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,2,2,2,2,2],[2]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,1],[2,2]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,2],[2,2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,2,2],[2,2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,2,2,2],[2,2]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1],[2,2,2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,2],[2,2,2]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[[1,1,1,1,1,1,3]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,1,2,3]]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,1,3,3]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,2,2,3]]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,2,3,3]]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,3,3,3]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,2,2,2,3]]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,2,2,3,3]]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,2,3,3,3]]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,3,3,3,3]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,2,2,2,3]]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,2,2,3,3]]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,2,3,3,3]]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3,3,3,3]]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3,3,3,3]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,2,2,2,3]]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,2,2,3,3]]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,2,3,3,3]]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3,3,3,3]]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3,3,3,3]]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3,3,3,3]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,2,2,2,3]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,2,2,3,3]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,2,3,3,3]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3,3,3,3]]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3,3,3,3]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3,3,3,3]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,1,1],[3]]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,1,2],[3]]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,1,3],[2]]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,1,3],[3]]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1,1,2,2],[3]]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}

Description

The number of circled entries of the shifted recording tableau of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the number of circled entries in

$Q$ .

The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001152The number of pairs with even minimum in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001401The number of distinct entries in a semistandard tableau. St000259The diameter of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001877Number of indecomposable injective modules with projective dimension 2. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000264The girth of a graph, which is not a tree. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.