Identifier
Values
[[3],[]] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[[3,1],[1]] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[[3,2],[2]] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[[1,1,1],[]] => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 3
[[3,2,1],[2,1]] => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 3
[[4],[]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[4,1],[1]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[2,2],[]] => [2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[4,2],[2]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[4,2,1],[2,1]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[4,3],[3]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[2,2,1],[1]] => [2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[4,3,1],[3,1]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[4,3,2],[3,2]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[1,1,1,1],[]] => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[2,2,1,1],[1,1]] => [2,2] => [[2,2],[]] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[4,3,2,1],[3,2,1]] => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 4
[[5],[]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,1],[1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,2],[2]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,2,1],[2,1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,3],[3]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,3,1],[3,1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,3,2],[3,2]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,3,2,1],[3,2,1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4],[4]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4,1],[4,1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4,2],[4,2]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4,2,1],[4,2,1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4,3],[4,3]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4,3,1],[4,3,1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4,3,2],[4,3,2]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[1,1,1,1,1],[]] => [1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[5,4,3,2,1],[4,3,2,1]] => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[6],[]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,1],[1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,2],[2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,2,1],[2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[3,3],[]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,3],[3]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,3,1],[3,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,3,2],[3,2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,3,2,1],[3,2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,4],[4]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[3,3,1],[1]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,4,1],[4,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[2,2,2],[]] => [2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,4,2],[4,2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[3,3,1,1],[1,1]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,4,2,1],[4,2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,4,3],[4,3]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,4,3,1],[4,3,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,4,3,2],[4,3,2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,4,3,2,1],[4,3,2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5],[5]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,1],[5,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[3,3,2],[2]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,5,2],[5,2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,2,1],[5,2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,3],[5,3]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[2,2,2,1],[1]] => [2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[3,3,2,1],[2,1]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,5,3,1],[5,3,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,3,2],[5,3,2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[3,3,2,1,1],[2,1,1]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,5,3,2,1],[5,3,2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,4],[5,4]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,4,1],[5,4,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[3,3,2,2],[2,2]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,5,4,2],[5,4,2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[2,2,2,1,1],[1,1]] => [2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,5,4,2,1],[5,4,2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,4,3],[5,4,3]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[3,3,2,2,1],[2,2,1]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,5,4,3,1],[5,4,3,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[6,5,4,3,2],[5,4,3,2]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[2,2,2,1,1,1],[1,1,1]] => [2,2,2] => [[2,2,2],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[3,3,2,2,1,1],[2,2,1,1]] => [3,3] => [[3,3],[]] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[6,5,4,3,2,1],[5,4,3,2,1]] => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[7],[]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,1],[1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,2],[2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,2,1],[2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,3],[3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,3,1],[3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,3,2],[3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,3,2,1],[3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4],[4]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4,1],[4,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4,2],[4,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4,2,1],[4,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4,3],[4,3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4,3,1],[4,3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4,3,2],[4,3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,4,3,2,1],[4,3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5],[5]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,1],[5,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,2],[5,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,2,1],[5,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,3],[5,3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
>>> Load all 145 entries. <<<
[[7,5,3,1],[5,3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,3,2],[5,3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,3,2,1],[5,3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4],[5,4]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4,1],[5,4,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4,2],[5,4,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4,2,1],[5,4,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4,3],[5,4,3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4,3,1],[5,4,3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4,3,2],[5,4,3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,5,4,3,2,1],[5,4,3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6],[6]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,1],[6,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,2],[6,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,2,1],[6,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,3],[6,3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,3,1],[6,3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,3,2],[6,3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,3,2,1],[6,3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4],[6,4]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4,1],[6,4,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4,2],[6,4,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4,2,1],[6,4,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4,3],[6,4,3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4,3,1],[6,4,3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4,3,2],[6,4,3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,4,3,2,1],[6,4,3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5],[6,5]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,1],[6,5,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,2],[6,5,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,2,1],[6,5,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,3],[6,5,3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,3,1],[6,5,3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,3,2],[6,5,3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,3,2,1],[6,5,3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4],[6,5,4]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4,1],[6,5,4,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4,2],[6,5,4,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4,2,1],[6,5,4,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4,3],[6,5,4,3]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4,3,1],[6,5,4,3,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4,3,2],[6,5,4,3,2]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[7,6,5,4,3,2,1],[6,5,4,3,2,1]] => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
search for individual values
searching the database for the individual values of this statistic
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Map
to skew partition
Description
The partition regarded as a skew partition.
Map
dominating partition
Description
The dominating partition in the Schur expansion.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a sublattice of the dominance order and that its top element is the conjugate of the partition formed by sorting the column lengths of $\lambda / \mu$ into decreasing order.
This map returns the largest partition $\nu$ in dominance order for which $c^\lambda_{\mu, \nu}$ is positive.
For example,
$$ s_{331/2} = s_{311} + s_{32}, $$
and the partition $32$ dominates $311$.
Map
cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.