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Matching statistic: St001176
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: 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],[2],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,1]
=> 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000319
Mp00077: Semistandard tableaux —shape⟶ 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%
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],[3]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[2],[4]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1],[2],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[3],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[4],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[3],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[4],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[3],[4],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[[1],[2],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[3],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[4],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[3],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[4],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[3],[4],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[3],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[4],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 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
Mp00077: Semistandard tableaux —shape⟶ 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%
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],[3]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[2],[4]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1],[2],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[3],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[4],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[3],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[4],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[3],[4],[5]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [4]
=> 3 = 2 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [3,1,1]
=> 2 = 1 + 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [3,2]
=> 2 = 1 + 1
[[1],[2],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[3],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[4],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[3],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[4],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[2],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[3],[4],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[3],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[4],[5],[6]]
=> [1,1,1]
=> [3]
=> 2 = 1 + 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 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: St000228
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: 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]
=> 2
[[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 size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000394
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000459
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000459: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000459: 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]
=> 2
[[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 hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.
Matching statistic: St000460
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000460: 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]
=> 2
[[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 hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St000870
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000870: 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]
=> 2
[[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 product of the hook lengths of the diagonal cells in an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below + 1. This statistic is the product of the hook lengths of the diagonal cells $(i,i)$ of a partition.
Matching statistic: St001480
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001480: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
Description
The number of simple summands of the module J^2/J^3. Here J is the Jacobson radical of the Nakayama algebra algebra corresponding to the Dyck path.
Matching statistic: St000738
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ 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],[2]]
=> 2 = 1 + 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
The following 160 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000744The length of the path to the largest entry in a standard Young tableau. St000157The number of descents of a standard tableau. St000203The number of external nodes of a binary tree. St000637The length of the longest cycle in a graph. St000673The number of non-fixed points of a permutation. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000007The number of saliances of the permutation. St000028The number of stack-sorts needed to sort a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000051The size of the left subtree of a binary tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000837The number of ascents of distance 2 of a permutation. St000990The first ascent of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St000314The number of left-to-right-maxima of a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. 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. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001330The hat guessing number of a graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St000508Eigenvalues of the random-to-random operator acting on a simple module. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001557The number of inversions of the second entry of a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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 $K[x]/(x^n)$. 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. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St001960The number of descents of a permutation minus one if its first entry is not one. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001896The number of right descents of a signed permutations. St001866The nesting alignments of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001644The dimension of a graph. St000717The number of ordinal summands of a poset. St001946The number of descents in a parking function. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001937The size of the center of a parking function. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. 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. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001904The length of the initial strictly increasing segment of a parking function. St000068The number of minimal elements in a poset. St000911The number of maximal antichains of maximal size in a poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001490The number of connected components of a skew partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001618The cardinality of the Frattini sublattice of a lattice. St001845The number of join irreducibles minus the rank 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. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension 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. St001301The first Betti number of the order complex associated with the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. 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. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000080The rank of the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001625The Möbius invariant of a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001890The maximum magnitude of the Möbius function of a poset. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000100The number of linear extensions of a poset. St000168The number of internal nodes of an ordered tree. St000307The number of rowmotion orbits of a poset. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. 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 $eA$. St001623The number of doubly irreducible elements of a lattice. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a 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. St001964The interval resolution global dimension of a poset. 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. St001621The number of atoms of a lattice. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000189The number of elements in the poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000635The number of strictly order preserving maps of a poset into itself. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001877Number of indecomposable injective modules with projective dimension 2. 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. 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 $K[x]/(x^n)$. 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. St000422The energy of a graph, if it is integral.
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