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Matching statistic: St000319
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 0
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [1]
=> 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [1]
=> 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [1]
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [1]
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 0
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [4,1]
=> [1]
=> 0
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [4,1]
=> [1]
=> 0
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1]
=> [1]
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [4,1]
=> [1]
=> 0
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [1]
=> 0
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,2]
=> [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
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 0
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [1]
=> 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [1]
=> 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [1]
=> 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [1]
=> 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [1]
=> 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [1]
=> 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 0
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [4,1]
=> [1]
=> 0
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [4,1]
=> [1]
=> 0
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1]
=> [1]
=> 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [4,1]
=> [1]
=> 0
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 1
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [1]
=> 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [1]
=> 0
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [2,2]
=> [2]
=> 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,2]
=> [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: St000147
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2]
=> 2 = 1 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [1]
=> 1 = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [1]
=> 1 = 0 + 1
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [4,1]
=> [1]
=> 1 = 0 + 1
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [4,1]
=> [1]
=> 1 = 0 + 1
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1]
=> [1]
=> 1 = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [4,1]
=> [1]
=> 1 = 0 + 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> 2 = 1 + 1
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [1]
=> 1 = 0 + 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [2,2]
=> [2]
=> 2 = 1 + 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [2,2]
=> [2]
=> 2 = 1 + 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [2,2]
=> [2]
=> 2 = 1 + 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,2]
=> [2]
=> 2 = 1 + 1
Description
The largest part of an integer partition.
Matching statistic: St001280
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [2,2]
=> [2,2]
=> 2 = 1 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [2,1]
=> [2,1]
=> 1 = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [4,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,2]
=> [2,2,1]
=> 2 = 1 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [3,2]
=> [2,2,1]
=> 2 = 1 + 1
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [2,2]
=> [2,2]
=> 2 = 1 + 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [2,2]
=> [2,2]
=> 2 = 1 + 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [2,2]
=> [2,2]
=> 2 = 1 + 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,2]
=> [2,2]
=> 2 = 1 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000451
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000451: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => [2,3,1] => 2 = 0 + 2
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => [1,3,2] => 2 = 0 + 2
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => 2 = 0 + 2
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [1,3,4,2] => 2 = 0 + 2
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => 2 = 0 + 2
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 3 = 1 + 2
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1,3] => [2,3,1] => 2 = 0 + 2
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1,3] => [2,3,1] => 2 = 0 + 2
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => [2,3,1] => 2 = 0 + 2
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [3,1,2] => [1,3,2] => 2 = 0 + 2
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2 = 0 + 2
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [3,1,2] => [1,3,2] => 2 = 0 + 2
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => [2,3,1] => 2 = 0 + 2
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => [1,3,2] => 2 = 0 + 2
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 2 = 0 + 2
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 2 = 0 + 2
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => 2 = 0 + 2
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,1,2,4,5] => [1,3,4,5,2] => 2 = 0 + 2
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1,2,3,5] => [1,2,4,5,3] => 2 = 0 + 2
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 2 = 0 + 2
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,5,1,2] => 3 = 1 + 2
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [1,4,5,2,3] => 3 = 1 + 2
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [2,1,3,4] => [2,3,4,1] => 2 = 0 + 2
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => [2,3,4,1] => 2 = 0 + 2
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => [2,3,4,1] => 2 = 0 + 2
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1,2,4] => [1,3,4,2] => 2 = 0 + 2
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => [1,3,4,2] => 2 = 0 + 2
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1,2,4] => [1,3,4,2] => 2 = 0 + 2
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => [2,3,4,1] => 2 = 0 + 2
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1,2,4] => [1,3,4,2] => 2 = 0 + 2
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => 2 = 0 + 2
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => 2 = 0 + 2
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => 2 = 0 + 2
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [4,1,2,3] => [1,2,4,3] => 2 = 0 + 2
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => [2,3,4,1] => 2 = 0 + 2
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [1,3,4,2] => 2 = 0 + 2
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => 2 = 0 + 2
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 3 = 1 + 2
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 3 = 1 + 2
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 3 = 1 + 2
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 3 = 1 + 2
[[4,3],[3]]
=> [[1,1,1,2],[2,2,2]]
=> [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => ? = 2 + 2
[[4,3,2,1],[4,3,2],[4,3],[4]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [1,5,8,10,2,6,9,3,7,4] => ? = 2 + 2
[[4,3,2,1],[4,3,2],[4,3],[3]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [4,5,8,10,1,6,9,2,7,3] => ? = 2 + 2
[[4,3,2,1],[4,3,2],[4,2],[4]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [1,7,8,10,2,5,9,3,6,4] => ? = 2 + 2
[[4,3,2,1],[4,3,2],[4,2],[3]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [1,7,8,10,4,5,9,2,6,3] => ? = 2 + 2
[[4,3,2,1],[4,3,2],[3,2],[3]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [6,7,8,10,1,4,9,2,5,3] => ? = 2 + 2
[[4,3,2,1],[4,3,2],[4,2],[2]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [3,7,8,10,4,5,9,1,6,2] => ? = 2 + 2
[[4,3,2,1],[4,3,2],[3,2],[2]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [6,7,8,10,3,4,9,1,5,2] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,3],[4]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [1,5,9,10,2,6,8,3,7,4] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,3],[3]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [4,5,9,10,1,6,8,2,7,3] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,2],[4]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [1,5,9,10,2,7,8,3,6,4] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,2],[3]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [1,4,9,10,2,7,8,5,6,3] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[3,2],[3]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [4,6,9,10,1,7,8,2,5,3] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,2],[2]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [3,4,9,10,1,7,8,5,6,2] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[4,2],[4]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [1,8,9,10,2,5,7,3,6,4] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[4,2],[3]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [1,8,9,10,4,5,7,2,6,3] => ? = 2 + 2
[[4,3,2,1],[3,2,1],[3,2],[3]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [7,8,9,10,1,4,6,2,5,3] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[4,2],[2]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [3,8,9,10,4,5,7,1,6,2] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[3,2],[2]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [3,8,9,10,4,6,7,1,5,2] => ? = 2 + 2
[[4,3,2,1],[3,2,1],[3,2],[2]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [7,8,9,10,3,4,6,1,5,2] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,1],[4]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [1,6,9,10,2,7,8,3,5,4] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,1],[3]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [1,6,9,10,2,7,8,4,5,3] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[3,1],[3]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [5,6,9,10,1,7,8,2,4,3] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,1],[2]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [1,6,9,10,3,7,8,4,5,2] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[3,1],[2]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [5,6,9,10,1,7,8,3,4,2] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[4,1],[4]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [1,8,9,10,2,6,7,3,5,4] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[4,1],[3]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [1,8,9,10,2,6,7,4,5,3] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[3,1],[3]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [1,8,9,10,5,6,7,2,4,3] => ? = 2 + 2
[[4,3,2,1],[3,2,1],[3,1],[3]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [7,8,9,10,1,5,6,2,4,3] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[4,1],[2]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => [1,8,9,10,3,6,7,4,5,2] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[3,1],[2]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => [1,8,9,10,5,6,7,3,4,2] => ? = 2 + 2
[[4,3,2,1],[3,2,1],[3,1],[2]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => [7,8,9,10,1,5,6,3,4,2] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[2,1],[2]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => [4,8,9,10,5,6,7,1,3,2] => ? = 2 + 2
[[4,3,2,1],[3,2,1],[2,1],[2]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => [7,8,9,10,4,5,6,1,3,2] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[4,1],[1]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => [2,6,9,10,3,7,8,4,5,1] => ? = 2 + 2
[[4,3,2,1],[4,3,1],[3,1],[1]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,2,6,7,1,3,4,8] => [5,6,9,10,2,7,8,3,4,1] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[4,1],[1]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => [2,8,9,10,3,6,7,4,5,1] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[3,1],[1]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => [2,8,9,10,5,6,7,3,4,1] => ? = 2 + 2
[[4,3,2,1],[3,2,1],[3,1],[1]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => [7,8,9,10,2,5,6,3,4,1] => ? = 2 + 2
[[4,3,2,1],[4,2,1],[2,1],[1]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => [4,8,9,10,5,6,7,2,3,1] => ? = 2 + 2
[[4,3,2,1],[3,2,1],[2,1],[1]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,2,5,9,1,3,6,10] => [7,8,9,10,4,5,6,2,3,1] => ? = 2 + 2
Description
The length of the longest pattern of the form k 1 2...(k-1).
Matching statistic: St001882
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 0
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => [2,1,3] => 0
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => [3,1,2] => 0
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => [2,1] => 0
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => [2,1] => 0
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 0
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => 0
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => 0
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1,3] => [2,1,3] => 0
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1,3] => [2,1,3] => 0
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 0
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [3,1,2] => [3,1,2] => 0
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 0
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [3,1,2] => [3,1,2] => 0
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => [2,1,3] => 0
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => [3,1,2] => 0
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 0
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => [2,1] => 0
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => [2,1] => 0
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => [2,1] => 0
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => [2,1] => 0
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => [2,1] => 0
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 0
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => 0
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => 0
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => 0
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => 0
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => 0
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1,2,4] => [3,1,2,4] => 0
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => 0
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => 0
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => 0
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [4,1,2,3] => [4,1,2,3] => 0
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 0
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => 0
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => 0
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 1
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 1
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 0
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 0
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 0
[[2,1,0,0],[1,0,0],[0,0],[0]]
=> [[3,4],[4]]
=> [2,1,3] => [2,1,3] => 0
[[5,1],[1]]
=> [[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 0
[[5,1],[2]]
=> [[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 0
[[5,1],[3]]
=> [[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 0
[[5,1],[4]]
=> [[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 0
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 0
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 1
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 1
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 1
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2
[[4,1,0],[1,0],[0]]
=> [[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[4,1,0],[1,0],[1]]
=> [[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[4,1,0],[1,1],[1]]
=> [[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[4,1,0],[2,0],[0]]
=> [[2,2,3,3],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[4,1,0],[2,0],[1]]
=> [[1,2,3,3],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[4,1,0],[2,1],[1]]
=> [[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[4,1,0],[3,0],[0]]
=> [[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[4,1,0],[3,0],[1]]
=> [[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[4,1,0],[3,0],[2]]
=> [[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[4,1,0],[3,0],[3]]
=> [[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[4,1,0],[3,1],[1]]
=> [[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[4,1,0],[3,1],[3]]
=> [[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[4,1,0],[4,0],[0]]
=> [[2,2,2,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[4,1,0],[4,0],[1]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[4,1,0],[4,0],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[4,1,0],[4,0],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[4,1,0],[4,0],[4]]
=> [[1,1,1,1],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[4,1,0],[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[4,1,0],[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[4,1,0],[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St001904
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001904: Parking functions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001904: Parking functions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => [2,1,3] => 1 = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => [3,1,2] => 1 = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1,3] => [2,1,3] => 1 = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1,3] => [2,1,3] => 1 = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1 = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [3,1,2] => [3,1,2] => 1 = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => [3,1,2] => 1 = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [3,1,2] => [3,1,2] => 1 = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => [2,1,3] => 1 = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => [3,1,2] => 1 = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 0 + 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 + 1
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => [4,1,2,3] => 1 = 0 + 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => 1 = 0 + 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => 1 = 0 + 1
[[2,1,0,0],[1,0,0],[0,0],[0]]
=> [[3,4],[4]]
=> [2,1,3] => [2,1,3] => 1 = 0 + 1
[[5,1],[1]]
=> [[1,2,2,2,2],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 0 + 1
[[5,1],[2]]
=> [[1,1,2,2,2],[2]]
=> [3,1,2,4,5,6] => [3,1,2,4,5,6] => ? = 0 + 1
[[5,1],[3]]
=> [[1,1,1,2,2],[2]]
=> [4,1,2,3,5,6] => [4,1,2,3,5,6] => ? = 0 + 1
[[5,1],[4]]
=> [[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 0 + 1
[[5,1],[5]]
=> [[1,1,1,1,1],[2]]
=> [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 0 + 1
[[4,2],[2]]
=> [[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 1 + 1
[[4,2],[3]]
=> [[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => ? = 1 + 1
[[4,2],[4]]
=> [[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 1 + 1
[[3,3],[3]]
=> [[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 2 + 1
[[4,1,0],[1,0],[0]]
=> [[2,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[[4,1,0],[1,0],[1]]
=> [[1,3,3,3],[3]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[[4,1,0],[1,1],[1]]
=> [[1,3,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[[4,1,0],[2,0],[0]]
=> [[2,2,3,3],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[[4,1,0],[2,0],[1]]
=> [[1,2,3,3],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[[4,1,0],[2,0],[2]]
=> [[1,1,3,3],[3]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[[4,1,0],[2,1],[1]]
=> [[1,2,3,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[[4,1,0],[2,1],[2]]
=> [[1,1,3,3],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[[4,1,0],[3,0],[0]]
=> [[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[[4,1,0],[3,0],[1]]
=> [[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[[4,1,0],[3,0],[2]]
=> [[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[[4,1,0],[3,0],[3]]
=> [[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[[4,1,0],[3,1],[1]]
=> [[1,2,2,3],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[[4,1,0],[3,1],[2]]
=> [[1,1,2,3],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[[4,1,0],[3,1],[3]]
=> [[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[[4,1,0],[4,0],[0]]
=> [[2,2,2,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[[4,1,0],[4,0],[1]]
=> [[1,2,2,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[[4,1,0],[4,0],[2]]
=> [[1,1,2,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[[4,1,0],[4,0],[3]]
=> [[1,1,1,2],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[[4,1,0],[4,0],[4]]
=> [[1,1,1,1],[3]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[[4,1,0],[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 0 + 1
[[4,1,0],[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 0 + 1
[[4,1,0],[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 0 + 1
[[4,1,0],[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 0 + 1
[[3,2,0],[2,0],[0]]
=> [[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 1
[[3,2,0],[2,0],[1]]
=> [[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 1
[[3,2,0],[2,0],[2]]
=> [[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 1
[[3,2,0],[2,1],[1]]
=> [[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 1 + 1
[[3,2,0],[2,1],[2]]
=> [[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 1
[[3,2,0],[2,2],[2]]
=> [[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1 + 1
[[3,2,0],[3,0],[0]]
=> [[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 + 1
[[3,2,0],[3,0],[1]]
=> [[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 + 1
[[3,2,0],[3,0],[2]]
=> [[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 + 1
[[3,2,0],[3,0],[3]]
=> [[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => ? = 1 + 1
[[3,2,0],[3,1],[1]]
=> [[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 1 + 1
Description
The length of the initial strictly increasing segment of a parking function.
Matching statistic: St001876
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
[[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
[[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0 + 1
[[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
[[2,1,0],[1,0],[0]]
=> [[2,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0],[1,0],[1]]
=> [[1,3],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0],[1,1],[1]]
=> [[1,3],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0],[2,0],[0]]
=> [[2,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0],[2,0],[1]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0],[2,0],[2]]
=> [[1,1],[3]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0],[2,1],[1]]
=> [[1,2],[2]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0],[2,1],[2]]
=> [[1,1],[2]]
=> [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[1,1,1],[1,1],[1]]
=> [[1],[2],[3]]
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0 + 1
[[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[4,1],[1]]
=> [[1,2,2,2],[2]]
=> [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 0 + 1
[[4,1],[2]]
=> [[1,1,2,2],[2]]
=> [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 0 + 1
[[4,1],[3]]
=> [[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 0 + 1
[[4,1],[4]]
=> [[1,1,1,1],[2]]
=> [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 0 + 1
[[3,2],[2]]
=> [[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 1 + 1
[[3,2],[3]]
=> [[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 1
[[3,1,0],[1,0],[0]]
=> [[2,3,3],[3]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
[[3,1,0],[1,0],[1]]
=> [[1,3,3],[3]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
[[3,1,0],[1,1],[1]]
=> [[1,3,3],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
[[3,1,0],[2,0],[0]]
=> [[2,2,3],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
[[3,1,0],[2,0],[1]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
[[3,1,0],[2,0],[2]]
=> [[1,1,3],[3]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
[[3,1,0],[2,1],[1]]
=> [[1,2,3],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
[[3,1,0],[2,1],[2]]
=> [[1,1,3],[2]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
[[3,1,0],[3,0],[0]]
=> [[2,2,2],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0 + 1
[[3,1,0],[3,0],[1]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0 + 1
[[3,1,0],[3,0],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0 + 1
[[3,1,0],[3,0],[3]]
=> [[1,1,1],[3]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0 + 1
[[3,1,0],[3,1],[1]]
=> [[1,2,2],[2]]
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
[[3,1,0],[3,1],[2]]
=> [[1,1,2],[2]]
=> [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
[[3,1,0],[3,1],[3]]
=> [[1,1,1],[2]]
=> [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0 + 1
[[2,2,0],[2,0],[0]]
=> [[2,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
[[2,2,0],[2,0],[1]]
=> [[1,2],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
[[2,2,0],[2,0],[2]]
=> [[1,1],[3,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
[[2,2,0],[2,1],[1]]
=> [[1,2],[2,3]]
=> [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 1 + 1
[[2,2,0],[2,1],[2]]
=> [[1,1],[2,3]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
[[2,2,0],[2,2],[2]]
=> [[1,1],[2,2]]
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1 + 1
[[2,1,1],[1,1],[1]]
=> [[1,3],[2],[3]]
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0 + 1
[[2,1,1],[2,1],[1]]
=> [[1,2],[2],[3]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0 + 1
[[2,1,1],[2,1],[2]]
=> [[1,1],[2],[3]]
=> [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0 + 1
[[2,1,0,0],[1,0,0],[0,0],[0]]
=> [[3,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0,0],[1,0,0],[1,0],[0]]
=> [[2,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0,0],[1,0,0],[1,0],[1]]
=> [[1,4],[4]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0,0],[1,1,0],[1,0],[0]]
=> [[2,4],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[2,1,0,0],[1,1,0],[1,0],[1]]
=> [[1,4],[3]]
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0 + 1
[[1,1,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,0,0,0,0],[0,0,0,0],[0,0,0],[0,0],[0]]
=> [[5],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[0,0],[0]]
=> [[3],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[0]]
=> [[2],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,0,0],[1,0],[1]]
=> [[1],[4]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[0]]
=> [[2],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,0],[1]]
=> [[1],[3]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]]
=> [[1],[2]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0],[0,0,0,0],[0,0,0],[0,0],[0]]
=> [[6],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[0,0,0,0],[0,0,0],[0,0],[0]]
=> [[5],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[7]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,0,0,0,0],[0,0,0,0],[0,0,0],[0,0],[0]]
=> [[5],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[6]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[0,0,0],[0,0],[0]]
=> [[4],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[0,0],[0]]
=> [[3],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[0]]
=> [[2],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]]
=> [[1],[5]]
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
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