Tied dice

Tied dice Kronenthal, B. G.; Traldi, Lorenzo A generalized die is a list (x1,…,xn) of integers. For integers n≥1, a≤b and s let D(n,a,b,s) be the set of all dice with a≤x1≤⋯≤xn≤b and ∑xi=s. Two dice X and Y are tied if the number of pairs (i,j) with xiyj. We prove the following: with one exception (unique up to isomorphism), if X≠Y∈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∈D(n,a,b,s) which ties neither X nor Y