素数幂骰子尺寸的二维骰子游戏数量
On the number of $2$-dice games with prime power dice-size
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中文总结 AI 辅助
该研究针对素数幂尺寸的Sicherman骰子，利用生成多项式技术给出了满足和分布条件的二维骰子游戏数量的上下界，并证明了对应的非闭合组合公式。
中文摘要 AI 辅助
我们有两枚各含n个标签的骰子，希望为它们分配一对正整数标签，使得两枚骰子上标签的和相同，且与标准标签的和分布一致，我们将这样的一对标签称为尺寸为2（骰子尺寸为m）的（骰子）游戏。这里我们针对骰子尺寸为素数幂p^k的Sicherman骰子，给出了尺寸为2的骰子游戏的下界和上界。我们利用生成多项式这一标准技术，证明了一个非闭合的组合公式。
英文摘要
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.
