Suppose $\lambda\vdash n$ is a partition. Associated with this partition is the set of Standard Young Tableau $\text{SYT}(\lambda)$ such that the associated Young Diagram is filled in with the numbers $\{1,\dots,n\}$, with rows and columns strictly increasing.
For example, if $\lambda=(3,2)$, the Young Tableaux of this shape are given by
$$\begin{matrix}1 & 2 & 3 \\ 4 & 5\end{matrix},
\quad\begin{matrix}1 & 2 & 4 \\ 3 & 5\end{matrix},
\quad\begin{matrix}1 & 3 & 4 \\ 2 & 5\end{matrix},
\quad\begin{matrix}1 & 2 & 5 \\ 3 & 4\end{matrix},
\quad\begin{matrix}1 & 3 & 5 \\ 2 & 4\end{matrix}$$
The element $i$ is in the descent set of a tableau if $i+1$ occurs strictly below $i$. Hence, the descent sets of the above are
$$\{3\},\quad\quad\{2,4\},\quad\quad\{1,4\},\quad\quad\{2\},\quad\quad\{1,3\}$$
and we can see that each of these are distinct.
My question then applies to the general case. For a given partition $\lambda\vdash n$, are the descent sets of the standard tableaux in $\text{SYT}(\lambda)$ dinstinct?
Best Answer
Consider the partition $\lambda=\langle 3,2,2,1\rangle$. It has standard Young tableaux
$$\begin{array}{ccc} 1&3&4\\ 2&7\\ 5&8\\ 6 \end{array}\qquad\text{and}\qquad \begin{array}{ccc} 1&3&4\\ 2&5\\ 6&7\\ 8 \end{array}$$
Both of these have descent set $\{1,4,5,7\}$.