# The Unapologetic Mathematician

## Generalized Young Tableaux

And now we have another generalization of Young tableaux. These are the same, except now we allow repetitions of the entries.

Explicitly, a generalized Young tableau $T$ — we write them with capital letters — of shape $\lambda$ is an array obtained by replacing the points of the Ferrers diagram of $\lambda$ with positive integers. Any skipped or repeated numbers are fine. We say that the “content” of $T$ is the composition $\mu=(\mu_1,\dots,\mu_m)$ where $\mu_i$ is the number of $i$ entries in $T$.

As an example, we have the generalized Young tableau

$\displaystyle\begin{array}{ccc}4&1&4\\1&3&\end{array}$

of shape $(3,2)$ and content $(2,0,1,2)$.

Notice that if $\lambda\vdash n$, then $\mu\vdash n$ as well, since both count up the total number of places in the tableau. Given a partition $\lambda$ and a composition $\mu$, both decomposing the same number $n$, we define $T_{\lambda\mu}$ to be the collection of generalized Young tableaux of shape $\lambda$ and content $\mu$. All the tableaux we’ve considered up until now have content $(1,\dots,1)=(1^n)$.

Now, pick some fixed (ungeneralized) tableau $t$. We can use the same one we usually do, numbering the rows from $1$ to $n$ across each row and from top to bottom, but it doesn’t really matter which we use. For our examples we’ll pick

$\displaystyle t=\begin{array}{ccc}1&2&3\\4&5&\end{array}$

Using this “reference” tableau, we can rewrite any generalized tableau as a function; define $T(i)$ to be the entry of $T$ in the same place as $i$ is in $t$. That is, any generalized tableau looks like

$\displaystyle\begin{array}{ccc}T(1)&T(2)&T(3)\\T(4)&T(5)&\end{array}$

and in our particular example above we have $T(1)=T(3)=4$, $T(2)=T(4)=1$, and $T(5)=3$. Conversely, any such function assigning a positive integer to each number from $1$ to $n$ can be interpreted as a generalized Young tableau. Of course the particular correspondence depends on exactly which reference tableau we use, but there will always be some such correspondence between functions and generalized tableaux.