Identifier
Mp00082:
Standard tableaux
—to Gelfand-Tsetlin pattern⟶
Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Images
=>
Cc0007;cc-rep-0Cc0018;cc-rep-1Cc0019;cc-rep-2Cc0002;cc-rep-3
[[1,2]]=>[[2,0],[1]]=>[[1,2]]=>[2]
[[1],[2]]=>[[1,1],[1]]=>[[1],[2]]=>[1,1]
[[1,2,3]]=>[[3,0,0],[2,0],[1]]=>[[1,2,3]]=>[3]
[[1,3],[2]]=>[[2,1,0],[1,1],[1]]=>[[1,3],[2]]=>[2,1]
[[1,2],[3]]=>[[2,1,0],[2,0],[1]]=>[[1,2],[3]]=>[2,1]
[[1],[2],[3]]=>[[1,1,1],[1,1],[1]]=>[[1],[2],[3]]=>[1,1,1]
[[1,2,3,4]]=>[[4,0,0,0],[3,0,0],[2,0],[1]]=>[[1,2,3,4]]=>[4]
[[1,3,4],[2]]=>[[3,1,0,0],[2,1,0],[1,1],[1]]=>[[1,3,4],[2]]=>[3,1]
[[1,2,4],[3]]=>[[3,1,0,0],[2,1,0],[2,0],[1]]=>[[1,2,4],[3]]=>[3,1]
[[1,2,3],[4]]=>[[3,1,0,0],[3,0,0],[2,0],[1]]=>[[1,2,3],[4]]=>[3,1]
[[1,3],[2,4]]=>[[2,2,0,0],[2,1,0],[1,1],[1]]=>[[1,3],[2,4]]=>[2,2]
[[1,2],[3,4]]=>[[2,2,0,0],[2,1,0],[2,0],[1]]=>[[1,2],[3,4]]=>[2,2]
[[1,4],[2],[3]]=>[[2,1,1,0],[1,1,1],[1,1],[1]]=>[[1,4],[2],[3]]=>[2,1,1]
[[1,3],[2],[4]]=>[[2,1,1,0],[2,1,0],[1,1],[1]]=>[[1,3],[2],[4]]=>[2,1,1]
[[1,2],[3],[4]]=>[[2,1,1,0],[2,1,0],[2,0],[1]]=>[[1,2],[3],[4]]=>[2,1,1]
[[1],[2],[3],[4]]=>[[1,1,1,1],[1,1,1],[1,1],[1]]=>[[1],[2],[3],[4]]=>[1,1,1,1]
[[1,2,3,4,5]]=>[[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]=>[[1,2,3,4,5]]=>[5]
[[1,3,4,5],[2]]=>[[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]=>[[1,3,4,5],[2]]=>[4,1]
[[1,2,4,5],[3]]=>[[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]=>[[1,2,4,5],[3]]=>[4,1]
[[1,2,3,5],[4]]=>[[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]=>[[1,2,3,5],[4]]=>[4,1]
[[1,2,3,4],[5]]=>[[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]=>[[1,2,3,4],[5]]=>[4,1]
[[1,3,5],[2,4]]=>[[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]=>[[1,3,5],[2,4]]=>[3,2]
[[1,2,5],[3,4]]=>[[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]=>[[1,2,5],[3,4]]=>[3,2]
[[1,3,4],[2,5]]=>[[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]=>[[1,3,4],[2,5]]=>[3,2]
[[1,2,4],[3,5]]=>[[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]=>[[1,2,4],[3,5]]=>[3,2]
[[1,2,3],[4,5]]=>[[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]=>[[1,2,3],[4,5]]=>[3,2]
[[1,4,5],[2],[3]]=>[[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]=>[[1,4,5],[2],[3]]=>[3,1,1]
[[1,3,5],[2],[4]]=>[[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]=>[[1,3,5],[2],[4]]=>[3,1,1]
[[1,2,5],[3],[4]]=>[[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]=>[[1,2,5],[3],[4]]=>[3,1,1]
[[1,3,4],[2],[5]]=>[[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]=>[[1,3,4],[2],[5]]=>[3,1,1]
[[1,2,4],[3],[5]]=>[[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]=>[[1,2,4],[3],[5]]=>[3,1,1]
[[1,2,3],[4],[5]]=>[[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]=>[[1,2,3],[4],[5]]=>[3,1,1]
[[1,4],[2,5],[3]]=>[[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]=>[[1,4],[2,5],[3]]=>[2,2,1]
[[1,3],[2,5],[4]]=>[[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]=>[[1,3],[2,5],[4]]=>[2,2,1]
[[1,2],[3,5],[4]]=>[[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]=>[[1,2],[3,5],[4]]=>[2,2,1]
[[1,3],[2,4],[5]]=>[[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]=>[[1,3],[2,4],[5]]=>[2,2,1]
[[1,2],[3,4],[5]]=>[[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]=>[[1,2],[3,4],[5]]=>[2,2,1]
[[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]]=>[2,1,1,1]
[[1,4],[2],[3],[5]]=>[[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]=>[[1,4],[2],[3],[5]]=>[2,1,1,1]
[[1,3],[2],[4],[5]]=>[[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]=>[[1,3],[2],[4],[5]]=>[2,1,1,1]
[[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]]=>[2,1,1,1]
[[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]]=>[1,1,1,1,1]
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.
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 $k=1,\dots,n$.
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
shape
Description
Return the shape of a tableau.
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