算术分拆和以及Z_n^k在对称群S_k作用下的轨道
数论
2007-05-23 v2 组合数学
摘要
我们研究M(n,k,r),即集合{(a_1,...,a_k)∈Z_n^k | a_1+...+a_k = r (mod n)}在S_k作用下的轨道数。等价地,M(n,k,r)对一算术序列的分拆数求和:M(n,k,r) = sum_{t ≥ 0} p(n-1,k,r+nt),其中p(a,b,t)表示将t分拆为至多b个部分且每一部分至多为a的分拆数。我们推导出此类算术分拆和的闭公式及各种恒等式。这些结果已出现于Elashvili/Jibladze/Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Alg. Combin. 10 (1999) 173-188,且主要结果亦由Von Sterneck发表于Sitzber. Akad. Wiss. Wien. Math. Naturw. Class. 111 (1902), 1567-1601(见math.NT/9909121中引理2及参考文献)。感谢Don Zagier与Robin Chapman提请我们注意这些文献。
引用
@article{arxiv.math/0106267,
title = {Arithmetic partition sums and orbits of Z_n^k under the symmetric group S_k},
author = {Matthias Beck and Alex J. Feingold and Michael D. Weiner},
journal= {arXiv preprint arXiv:math/0106267},
year = {2007}
}
备注
This paper has been withdrawn by the authors because the results have already appeared elsewhere