Tied dice

A \emph{generalized die} is a list \( (x_1,\ldots,x_n) \) of integers. For integers \( n \geq 1, a \leq b \) and \( s \), let \( D(n,a,b,s) \) be the set of all dice with \( a \leq x_1 \leq \ldots \leq x_n \leq b \) 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 \( X \neq Y \in D(n,a,b,s) \) 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 \( Y \).