Sylvester Denumerant的系数
Coefficients of Sylvester's Denumerant
摘要：
对于给定的$N+1$正整数序列$\mathbfα = [α_1,α_2,\dots,α_{N+1}]$，我们考虑了计算方程$α_1x_1+α_2 x_2+\cdots+α_{N} x_{N}+α_{N+1}x_{N+1}=t$的非负整数解的组合函数$E(\mathbfα)(t)$，其中右边的$t$是一个变化的非负整数.众所周知，$E(\mathbfα)(t)$是变量$t$的准多项式函数，次数为$N$。在组合数论中，这个函数被称为西尔维斯特的denumerant。我们的主要结果是一个新的算法，对于每一个固定的数$k$，计算在多项式时间的准多项式$E(\mathbfα)(t)$的最高$k+1$系数的步骤多项式的$t$（一个更简单，更明确的表示）。我们的算法是一个很好的偏序集结构上的极点相关的有理生成函数的$E(\mathbfα)(t)$和几何重新解释的一些有理生成函数在多面体锥的格点。我们的算法还使用Barvinok的基本快速分解的多面体锥到单模锥。本文还提出了一个简单的算法来预测第一个非常数系数，并得出结论，几个计算实验使用我们的算法在Latte积分的实施报告。我们比较它与各种Maple程序的部分或全部计算的denumerant。
英文摘要：
For a given sequence $\mathbfα = [α_1,α_2,\dots,α_{N+1}]$ of $N+1$ positive integers, we consider the combinatorial function $E(\mathbfα)(t)$ that counts the nonnegative integer solutions of the equation $α_1x_1+α_2 x_2+\cdots+α_{N} x_{N}+α_{N+1}x_{N+1}=t$, where the right-hand side $t$ is a varying nonnegative integer. It is well-known that $E(\mathbfα)(t)$ is a quasi-polynomial function in the variable $t$ of degree $N$. In combinatorial number theory this function is known as Sylvester's denumerant. Our main result is a new algorithm that, for every fixed number $k$, computes in polynomial time the highest $k+1$ coefficients of the quasi-polynomial $E(\mathbfα)(t)$ as step polynomials of $t$ (a simpler and more explicit representation). Our algorithm is a consequence of a nice poset structure on the poles of the associated rational generating function for $E(\mathbfα)(t)$ and the geometric reinterpretation of some rational generating functions in terms of lattice points in polyhedral cones. Our algorithm also uses Barvinok's fundamental fast decomposition of a polyhedral cone into unimodular cones. This paper also presents a simple algorithm to predict the first non-constant coefficient and concludes with a report of several computational experiments using an implementation of our algorithm in LattE integrale. We compare it with various Maple programs for partial or full computation of the denumerant.
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