A partition identity involving selected Ferrers-diagram points

Let

$$be an indeterminate. A partition$$

has size $||$ and length $l$; write $m_i$ for the multiplicity of the part $i$, and set

$$z_{}=_{i e 1} i^{m_i}m_i!.$$

For $r e 0$, let

denote the number of ways to choose $r$ points in the Ferrers diagram of

, with at least one selected point in every row. For an indeterminate $X$ and $n e 0$, write $(a)_s=a(a+1)(a+s-1)$ and $\binom{X}{n}=[X]_n/n!$, where $[X]_n=X(X-1)(X-n+1)$.

The partition identity conjecture. For all integers $n,r,s e 0$, the following equivalent identities hold:

_{:||=n}(-1)^{r-l}^{l-1}_i_s=(s-1)!. _{:||=n}^{l-1}_i_s=(s-1)!.

Here the two forms are related by replacing $X$ with $-X$.

The conjecture generalizes classical binomial identities expressed as sums over partitions. It is trivial for $r>n$, and the authors verified it for $n\leq7$, for $s=1$, and for $r=1,2,3$; its general case remains open.