Identifier
Values
[[1]] => [[1]] => [[1]] => ([],1) => 1
[[1,2]] => [[2,0],[1]] => [[1,2]] => ([(0,1)],2) => 1
[[1],[2]] => [[1,1],[1]] => [[1],[2]] => ([],1) => 1
[[1,2,3]] => [[3,0,0],[2,0],[1]] => [[1,2,3]] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 1
[[1,3],[2]] => [[2,1,0],[1,1],[1]] => [[1,3],[2]] => ([(0,2),(2,1)],3) => 1
[[1,2],[3]] => [[2,1,0],[2,0],[1]] => [[1,2],[3]] => ([(0,2),(2,1)],3) => 1
[[1],[2],[3]] => [[1,1,1],[1,1],[1]] => [[1],[2],[3]] => ([],1) => 1
[[1,4],[2],[3]] => [[2,1,1,0],[1,1,1],[1,1],[1]] => [[1,4],[2],[3]] => ([(0,3),(2,1),(3,2)],4) => 1
[[1,3],[2],[4]] => [[2,1,1,0],[2,1,0],[1,1],[1]] => [[1,3],[2],[4]] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 1
[[1,2],[3],[4]] => [[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3],[4]] => ([(0,3),(2,1),(3,2)],4) => 1
[[1],[2],[3],[4]] => [[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4]] => ([],1) => 1
[[1,5],[2],[3],[4]] => [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1,5],[2],[3],[4]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[1,2],[3],[4],[5]] => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]] => [[1,2],[3],[4],[5]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[[1],[2],[3],[4],[5]] => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]] => [[1],[2],[3],[4],[5]] => ([],1) => 1
[[1],[2],[3],[4],[5],[6]] => [[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
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The number of connected components of the Hasse diagram for the poset.
Map
to Gelfand-Tsetlin pattern
Description
Sends a tableau to its corresponding Gelfand-Tsetlin pattern.
To obtain this Gelfand-Tsetlin pattern, fill in the first row of the pattern with the shape of the tableau.
Then remove the maximal entry from the tableau to obtain a smaller tableau, and repeat the process until the tableau is empty.
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.