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- W4297923757 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." @default.
- W4297923757 created "2022-10-01" @default.
- W4297923757 creator A5076835963 @default.
- W4297923757 date "2015-12-13" @default.
- W4297923757 modified "2023-09-27" @default.
- W4297923757 title "A Probabilistic Characterization of the Dominance Order on Partitions" @default.
- W4297923757 doi "https://doi.org/10.48550/arxiv.1512.04084" @default.
- W4297923757 hasPublicationYear "2015" @default.
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