## What are these operations on ordinals, analogous to multichoose, called? [xpost from Math.SE]

Given well-ordered sets $\alpha$ and $\beta$, define $\left({\alpha\choose \beta}\right)$ to be the set of weakly decreasing functions from $\beta$ to $\alpha$, ordered lexicographically; this is in fact a well-ordering, so we get an operation on ordinals.

(I denote it by multichoose since for $n$, $k$ finite, $\left({n\choose k}\right)$ is indeed $\left({n\choose k}\right)$ in the usual sense.)

We can also order reverse-lexicographically and get a well-order; I'm calling that $\left({\alpha\choose \beta}\right)'$. Same questions about that. (And we can restrict to strictly decreasing functions to get $\binom{\alpha}{k}$ and $\binom{\alpha}{k}'$, but obviously $k$ has to be finite for that to be interesting.)