A Probabilistic Characterization of the Dominance Order on Partitions

TL;DR

This paper provides a probabilistic characterization of the dominance order on partitions using random variables and extends previous work, also exploring total non-negativity of certain matrices related to generating functions.

Contribution

It introduces a probabilistic criterion for dominance order on partitions and links it to total non-negativity properties of associated matrices, extending prior combinatorial results.

Findings

Dominance order characterized by probabilities of ball distributions in Ferrers diagrams.

Total non-negativity of matrices ${\\cal T}_p$ implies the same for ${\cal S}_p$.

A key step in the proof relies on the case $k=2$, connected to a potential combinatorial conjecture.

Abstract

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 $p \in (0,1)$. Fill each cell of $F$ 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. $\sum_{i=1}^k \nu(i) \geq \sum_{i=1}^k \mu(i)$ for all positive integers $k$, if and only if $P(\nu,j,t) \leq P(\mu,j,t)$ 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 $p = \{p_n\}_{n=0}^\infty$ be a sequence of real numbers. Let ${\cal T}_p$ be the $\mathbb{N}$ by $\mathbb{N}$ matrix with $({\cal T}_p)_{i,j} = p_{j-i}$ for all $i, j \in \mathbb{N}$ where we take $p_n = 0$ for $n < 0$. Let $(p^i)_j$ be the coefficient of $x^j$ in $(p(x))^i$ where $p(x) = \sum_{n=0}^\infty p_n x^n$ and $p^0(x) =1$. Let ${\cal S}_p$ be the $\mathbb{N}$ by $\mathbb{N}$ matrix with $({\cal S}_p)_{i,j} = (p^i)_j$ for all $i, j \in \mathbb{N}$. We show that if ${\cal T}_p$ is totally non-negative of order $k$ then so is ${\cal S}_p$. 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.