Identifier
Values
[[1,0],[1]] => [[1]] => ([],1) => ([],1) => 1
[[2,0],[2]] => [[1,1]] => ([],1) => ([],1) => 1
[[1,1],[1]] => [[1],[2]] => ([],1) => ([],1) => 1
[[1,0,0],[1,0],[1]] => [[1]] => ([],1) => ([],1) => 1
[[3,0],[3]] => [[1,1,1]] => ([],1) => ([],1) => 1
[[2,1],[2]] => [[1,1],[2]] => ([],1) => ([],1) => 1
[[2,0,0],[2,0],[2]] => [[1,1]] => ([],1) => ([],1) => 1
[[1,1,0],[1,1],[1]] => [[1],[2]] => ([],1) => ([],1) => 1
[[1,0,0,0],[1,0,0],[1,0],[1]] => [[1]] => ([],1) => ([],1) => 1
[[4,0],[4]] => [[1,1,1,1]] => ([],1) => ([],1) => 1
[[3,1],[3]] => [[1,1,1],[2]] => ([],1) => ([],1) => 1
[[2,2],[2]] => [[1,1],[2,2]] => ([],1) => ([],1) => 1
[[3,0,0],[3,0],[3]] => [[1,1,1]] => ([],1) => ([],1) => 1
[[2,1,0],[2,1],[2]] => [[1,1],[2]] => ([],1) => ([],1) => 1
[[1,1,1],[1,1],[1]] => [[1],[2],[3]] => ([],1) => ([],1) => 1
[[2,0,0,0],[2,0,0],[2,0],[2]] => [[1,1]] => ([],1) => ([],1) => 1
[[1,1,0,0],[1,1,0],[1,1],[1]] => [[1],[2]] => ([],1) => ([],1) => 1
[[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]] => [[1]] => ([],1) => ([],1) => 1
[[5,0],[5]] => [[1,1,1,1,1]] => ([],1) => ([],1) => 1
[[4,1],[4]] => [[1,1,1,1],[2]] => ([],1) => ([],1) => 1
[[3,2],[3]] => [[1,1,1],[2,2]] => ([],1) => ([],1) => 1
[[4,0,0],[4,0],[4]] => [[1,1,1,1]] => ([],1) => ([],1) => 1
[[3,1,0],[3,1],[3]] => [[1,1,1],[2]] => ([],1) => ([],1) => 1
[[2,2,0],[2,2],[2]] => [[1,1],[2,2]] => ([],1) => ([],1) => 1
[[2,1,1],[2,1],[2]] => [[1,1],[2],[3]] => ([],1) => ([],1) => 1
[[3,0,0,0],[3,0,0],[3,0],[3]] => [[1,1,1]] => ([],1) => ([],1) => 1
[[2,1,0,0],[2,1,0],[2,1],[2]] => [[1,1],[2]] => ([],1) => ([],1) => 1
[[1,1,1,0],[1,1,1],[1,1],[1]] => [[1],[2],[3]] => ([],1) => ([],1) => 1
[[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]] => [[1,1]] => ([],1) => ([],1) => 1
[[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]] => [[1],[2]] => ([],1) => ([],1) => 1
[[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]] => [[1]] => ([],1) => ([],1) => 1
[[5,1],[5]] => [[1,1,1,1,1],[2]] => ([],1) => ([],1) => 1
[[4,2],[4]] => [[1,1,1,1],[2,2]] => ([],1) => ([],1) => 1
[[3,3],[3]] => [[1,1,1],[2,2,2]] => ([],1) => ([],1) => 1
[[5,0,0],[5,0],[5]] => [[1,1,1,1,1]] => ([],1) => ([],1) => 1
[[4,1,0],[4,1],[4]] => [[1,1,1,1],[2]] => ([],1) => ([],1) => 1
[[3,2,0],[3,2],[3]] => [[1,1,1],[2,2]] => ([],1) => ([],1) => 1
[[3,1,1],[3,1],[3]] => [[1,1,1],[2],[3]] => ([],1) => ([],1) => 1
[[2,2,1],[2,2],[2]] => [[1,1],[2,2],[3]] => ([],1) => ([],1) => 1
[[4,0,0,0],[4,0,0],[4,0],[4]] => [[1,1,1,1]] => ([],1) => ([],1) => 1
[[3,1,0,0],[3,1,0],[3,1],[3]] => [[1,1,1],[2]] => ([],1) => ([],1) => 1
[[2,2,0,0],[2,2,0],[2,2],[2]] => [[1,1],[2,2]] => ([],1) => ([],1) => 1
[[2,1,1,0],[2,1,1],[2,1],[2]] => [[1,1],[2],[3]] => ([],1) => ([],1) => 1
[[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4]] => ([],1) => ([],1) => 1
[[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]] => [[1,1,1]] => ([],1) => ([],1) => 1
[[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]] => [[1,1],[2]] => ([],1) => ([],1) => 1
[[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]] => [[1],[2],[3]] => ([],1) => ([],1) => 1
[[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]] => [[1,1]] => ([],1) => ([],1) => 1
[[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]] => [[1],[2]] => ([],1) => ([],1) => 1
[[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]] => [[1]] => ([],1) => ([],1) => 1
[[6,1],[6]] => [[1,1,1,1,1,1],[2]] => ([],1) => ([],1) => 1
[[5,2],[5]] => [[1,1,1,1,1],[2,2]] => ([],1) => ([],1) => 1
[[4,3],[4]] => [[1,1,1,1],[2,2,2]] => ([],1) => ([],1) => 1
[[5,1,0],[5,1],[5]] => [[1,1,1,1,1],[2]] => ([],1) => ([],1) => 1
[[4,2,0],[4,2],[4]] => [[1,1,1,1],[2,2]] => ([],1) => ([],1) => 1
[[4,1,1],[4,1],[4]] => [[1,1,1,1],[2],[3]] => ([],1) => ([],1) => 1
[[3,3,0],[3,3],[3]] => [[1,1,1],[2,2,2]] => ([],1) => ([],1) => 1
[[3,2,1],[3,2],[3]] => [[1,1,1],[2,2],[3]] => ([],1) => ([],1) => 1
[[2,2,2],[2,2],[2]] => [[1,1],[2,2],[3,3]] => ([],1) => ([],1) => 1
[[5,0,0,0],[5,0,0],[5,0],[5]] => [[1,1,1,1,1]] => ([],1) => ([],1) => 1
[[4,1,0,0],[4,1,0],[4,1],[4]] => [[1,1,1,1],[2]] => ([],1) => ([],1) => 1
[[3,2,0,0],[3,2,0],[3,2],[3]] => [[1,1,1],[2,2]] => ([],1) => ([],1) => 1
[[3,1,1,0],[3,1,1],[3,1],[3]] => [[1,1,1],[2],[3]] => ([],1) => ([],1) => 1
[[2,2,1,0],[2,2,1],[2,2],[2]] => [[1,1],[2,2],[3]] => ([],1) => ([],1) => 1
[[2,1,1,1],[2,1,1],[2,1],[2]] => [[1,1],[2],[3],[4]] => ([],1) => ([],1) => 1
[[4,0,0,0,0],[4,0,0,0],[4,0,0],[4,0],[4]] => [[1,1,1,1]] => ([],1) => ([],1) => 1
[[3,1,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]] => [[1,1,1],[2]] => ([],1) => ([],1) => 1
[[2,2,0,0,0],[2,2,0,0],[2,2,0],[2,2],[2]] => [[1,1],[2,2]] => ([],1) => ([],1) => 1
[[2,1,1,0,0],[2,1,1,0],[2,1,1],[2,1],[2]] => [[1,1],[2],[3]] => ([],1) => ([],1) => 1
[[1,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4]] => ([],1) => ([],1) => 1
[[3,0,0,0,0,0],[3,0,0,0,0],[3,0,0,0],[3,0,0],[3,0],[3]] => [[1,1,1]] => ([],1) => ([],1) => 1
[[2,1,0,0,0,0],[2,1,0,0,0],[2,1,0,0],[2,1,0],[2,1],[2]] => [[1,1],[2]] => ([],1) => ([],1) => 1
[[1,1,1,0,0,0],[1,1,1,0,0],[1,1,1,0],[1,1,1],[1,1],[1]] => [[1],[2],[3]] => ([],1) => ([],1) => 1
[[2,0,0,0,0,0,0],[2,0,0,0,0,0],[2,0,0,0,0],[2,0,0,0],[2,0,0],[2,0],[2]] => [[1,1]] => ([],1) => ([],1) => 1
[[1,1,0,0,0,0,0],[1,1,0,0,0,0],[1,1,0,0,0],[1,1,0,0],[1,1,0],[1,1],[1]] => [[1],[2]] => ([],1) => ([],1) => 1
[[1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0],[1,0,0,0,0,0],[1,0,0,0,0],[1,0,0,0],[1,0,0],[1,0],[1]] => [[1]] => ([],1) => ([],1) => 1
[[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4],[5]] => ([],1) => ([],1) => 1
[[1]] => [[1]] => ([],1) => ([],1) => 1
[[1,1,1,1,1,1],[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4],[5],[6]] => ([],1) => ([],1) => 1
[[2]] => [[1,1]] => ([],1) => ([],1) => 1
[[3]] => [[1,1,1]] => ([],1) => ([],1) => 1
[[4]] => [[1,1,1,1]] => ([],1) => ([],1) => 1
[[5]] => [[1,1,1,1,1]] => ([],1) => ([],1) => 1
[[4,3,2,1],[4,3,2],[4,3],[4]] => [[1,1,1,1],[2,2,2],[3,3],[4]] => ([],1) => ([],1) => 1
search for individual values
searching the database for the individual values of this statistic
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Map
to semistandard tableau
Description
Return the Gelfand-Tsetlin pattern as a semistandard Young tableau.
Let $G$ be a Gelfand-Tsetlin pattern and let $\lambda^{(k)}$ be its $(n-k+1)$-st row. The defining inequalities of a Gelfand-Tsetlin pattern imply, regarding each row as a partition,
$$ \lambda^{(0)} \subseteq \lambda^{(1)} \subseteq \cdots \subseteq \lambda^{(n)}, $$
where $\lambda^{(0)}$ is the empty partition.
Each skew shape $\lambda^{(k)} / \lambda^{(k-1)}$ is moreover a horizontal strip.
We now define a semistandard tableau $T(G)$ by inserting $k$ into the cells of the skew shape $\lambda^{(k)} / \lambda^{(k-1)}$, for $k=1,\dots,n$.
Map
subcrystal
Description
The underlying poset of the subcrystal obtained by applying the raising operators to a semistandard tableau.
Map
incomparability graph
Description
The incomparability graph of a poset.