Tied dice

Abstract

A \emph{generalized die} is a list $(x_1,\dots ,x_n)$ of integers. For integers $n\ge 1,a\le b$ and $s$, let $D(n,a,b,s)$ be the set of all dice with $a\le x_1\le \dots \le x_n\le b$ and $\sum x_i=s$. Two dice $X$ and $Y$ are \emph{tied} if the number of pairs $(i,j)$ with $x_i{y}_j$. We prove the following: with one exception (unique up to isomorphism), if $X\ne Y\in D(n,a,b,s)$ are tied dice neither of which ties all other elements of $D(n,a,b,s)$, then there is a third die $Z\in D(n,a,b,s)$ which ties neither $X$ nor $Y$.