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 — we write them with capital letters — of shape is an array obtained by replacing the points of the Ferrers diagram of with positive integers. Any skipped or repeated numbers are fine. We say that the “content” of is the composition where is the number of entries in .
As an example, we have the generalized Young tableau
of shape and content .
Notice that if , then as well, since both count up the total number of places in the tableau. Given a partition and a composition , both decomposing the same number , we define to be the collection of generalized Young tableaux of shape and content . All the tableaux we’ve considered up until now have content .
Now, pick some fixed (ungeneralized) tableau . We can use the same one we usually do, numbering the rows from to across each row and from top to bottom, but it doesn’t really matter which we use. For our examples we’ll pick
Using this “reference” tableau, we can rewrite any generalized tableau as a function; define to be the entry of in the same place as is in . That is, any generalized tableau looks like
and in our particular example above we have , , and . Conversely, any such function assigning a positive integer to each number from to 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.