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We investigate the rank-generating function $F_{\lambda}$ of the poset ofpartitions contained inside a given shifted Ferrers shape $\lambda$. When$\lambda $ has four parts, we show that $F_{\lambda}$ is unimodal when $\lambda=\langle n,n-1,n-2,n-3 \rangle$, for any $n\ge 4$, and that unimodality failsfor the doubly-indexed, infinite family of partitions of the form$\lambda=\langle n,n-t,n-2t,n-3t \rangle$, for any given $t\ge 2$ and $n$ largeenough with respect to $t$.  When $\lambda $ has $b\le 3$ parts, we show that our rank-generatingfunctions $F_{\lambda}$ are all unimodal. However, the situation remains mostlyobscure for $b\ge 5$. In general, the type of results that we obtain presentsome remarkable similarities with those of the 1990 paper of D. Stanton, whoconsidered the case of partitions inside ordinary (straight) Ferrers shapes.  Along the way, we also determine some interesting $q$-analogs of the binomialcoefficients, which in certain instances we conjecture to be unimodal. We stateseveral other conjectures throughout this note, in the hopes to stimulatefurther work in this area. In particular, one of these will attempt to placeinto a much broader context the unimodality of the posets $M(n)$ of staircasepartitions, for which determining a combinatorial proof remains an outstandingopen problem.

Unimodality of partitions with distinct parts inside Ferrers shapes