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.

Applies the Schuetzenberger involution to a Gelfand-Tsetlin pattern.
The Schuetzenberger involution is usually regarded as an involution on semistandard Young tableaux with a fixed bound on the size of the entries. It is also known as evacuation, and in the context of crystal graphs of type $A$ it realizes Lusztig's involution.
In the language of tableaux it is defined as follows. Consider a semistandard tableau with no entry larger than $n$. Use Schuetzenberger's jeu de taquin to slide all entries equal to $1$ to the outer border of the tableau. Do the same for all entries equal to $2$, restricting the tableau to the entries larger than $2$, and so on, until the tableau is a reverse semistandard tableau. Finally, replace each entry with its complement with respect to $n$, that is, replace $e$ with $n+1-e$.

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 $k=1,\dots,n$.

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottommost row (in English notation).