Show that any number of partitions of $2r + k$ into $r + k$ parts is the same for any $k$.
I've tried this, but I haven't come up with anything; hence why I have nothing written here. But in any case, any sort of help is appreciated.
2026-04-01 01:34:59.1775007299
Combinatorial proof involving partitions and generating functions
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Make a Ferrers diagram of a partition of $2r+k$ into $r+k$ parts and remove the first column; what’s left is a partition of $r$, and every partition of $r$ can be extended uniquely to a partition of $2r+k$ by adding a column of $r+k$ dots at the left of the Ferrers diagram.