I've come across a function from the set of integer partitions to the natural numbers which I don't recognise but which probably ought to be familiar; it arises in the homogeneous Garnir relations for graded Specht modules (see Kleshchev, Mathas & Ram: "Universal graded Specht modules", Proc. LMS 105). I hope someone will recognise it and point me to a useful reference.

The function (let's call it $f$) is defined recursively; to begin with, $f(\varnothing)=1$. Now suppose we have a non-empty partition $\lambda$, and take a box $(i,j)$ in the Young diagram for which $i+j$ is maximised. Let $\mu$ be the partition you get by removing the first $i$ rows from the Young diagram and let $\nu$ be the partition you get by removing the first $j$ columns from the Young diagram. Now recursively define

$f(\lambda) = \binom{i+j}if(\mu)f(\nu)$.

For example, take $\lambda=(5,3,3,1)$. Let $(i,j)=(3,3)$, so that $\mu=(1)$ and $\nu=(2)$. One can calculate $f(\mu)=2$ and $f(\nu)=3$, so that $f(\lambda)=120$.

It's not hard to show that $f$ is well-defined (i.e. doesn't depend on the choice of $(i,j)$). I have two other definitions of $f$ (also recursive) which I'll omit for now (it's non-trivial to show that they are equivalent).

Please let me know if you've seen this function (or a similar-looking one) before. Since it's a function from partitions to positive integers, it ought to count tableaux of some kind.

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