A Probabilistic Characterization of the Dominance Order on Partitions

A probabilistic characterization of the dominance partial order on the set of partitions is presented. This extends work in “Symmetric polynomials and symmetric mean inequalities”. Electron. J. Combin., 20(3): Paper 34, 2013. Let $n$ be a positive integer and let $\nu$ be a partition of $n$. Let $F$ be the Ferrers diagram of $\nu$. Let $m$ be a positive integer and let $p \in (0,1)$\mathrm{.}Filleachcellof$F$ with balls, the number of which is independently drawn from the random variable $X=Bin(m,p)$. Given non-negative integers $j$ and $t$, let $P(\nu ,j,t)$ be the probability that the total number of balls in $F$ is $j$ and that no row of $F$ contains more that $t$ balls. We show that if $\nu$ and $\mu$ are partitions of $n$, then $\nu$ dominates $\mu$, i.e. ${\sum }_{i=1}^k\nu (i)\ge {\sum }_{i=1}^k\mu (i)$ for all positive integers $k$, if and only if $P(\nu ,j,t)\le P(\mu ,j,t)$ for all non-negative integers $j$ and $t$. It is also shown that this same result holds when $X$ is replaced by any one member of a large class of random variables. Let $p=\{p_n{\}}_{n=0}^{\mathrm{\infty }}$ be a sequence of real numbers. Let ${\cal T}p$bethe$$\mathbb{N}$$by$$\mathbb{N}$$matrixwith$({\cal T}p){i,j} = p{j-i}$forall$i, j \in $\mathbb{N}$$wherewetake$p_n = 0$for$n < 0$\mathrm{.}Let$(p^i)j$bethecoefficientof$x^j$in$(p(x))^i$where$p(x) = \sum{n=0}^\infty p_n x^n$and$p^0(x) =1$\mathrm{.}Let${\cal S}_p$bethe$$\mathbb{N}$$by$$\mathbb{N}$$matrixwith$({\cal S}p){i,j} = (p^i)_j$forall$i, j \in $\mathbb{N}$$\mathrm{.}Weshowthatif${\cal T}_p$ is totally non-negative of order $k$ then so is ${\mathcal{S}}_p$. The case $k=2$ of this result is a key step in the proof of the result on domination. We also show that the case $k=2$ would follow from a combinatorial conjecture that might be of independent interest.