In the paper under review, the author presents a probabilistic characterization of the dominance order on partitions.
Let \(\mathcal{LC}_r\) be the class of all log-concave integer valued random variables with \(\{i: P(X=i)>0\}=\{0,1,\ldots,r\}\). For example, given \(r\ge 1\), the binomial random variable \(\mathrm{Bin}(r,p)\) with \(p\in(0,1)\) is in \(\mathcal{LC}_r\).
Let \(Y_\nu\) be the Ferrers diagram of a partition \(\nu\). Assuming that the cells of \(Y_\nu\) are filled with independent and identically distributed draws from the random variable \(X\), then for \(j, t\ge 0\), let \(P(\nu,j,t)\) be the probability that the sum of all the entries in \(Y_\nu\) is \(j\) and the sum of the entries in each row of \(Y_\nu\) is \(\le t\).
Let \(\mu\) and \(\nu\) be two partitions of \(n\ge 0\). The author shows that if \(X\) is in \(\mathcal{LC}_r\) with \(r\ge 1\), then \(\nu\) dominates \(\mu\) (i.e. the Ferrers diagram of \(\mu\) is contained in the Ferrers diagram of \(\nu\)) if and only if \(P(\nu,j,t)\le P(\mu,j,t)\) for all \(j, t\ge 0\). The author also shows that this result can be connected with a more general result via Pólya frequency sequences.