On the number of $2$-dice games with prime power dice-size

Source abstract

We have two dices which have $n$ labels. We want to assign a pair of labelings with positive integers to them such that the sums of the labels on them are the same and have the same distribution as at the standard labeling. We call such a pair of labelings a (dice) game of size $2$ (with dice size $m$). Here we give a lower and an upper bound for the dice games of size $2$ with Sicherman dice of dice size $p^{k}$ where $p$ is a prime. We use the standard technique of generating polynomials to prove a non-closed combinatorial formula.