Arithmetic partition sums and orbits of Z_n^k under the symmetric group S_k

2001/06/29 by Matthias Beck, Beck, Matthias, Alex J. Feingold +3
Mathematics · #11B65 (Secondary) #11P81 #5A17 (Primary) 5A15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:11B65 #msc:11P81 #msc:5A15 #msc:5A17

paper · pdf · doi:10.48550/arxiv.math/0106267

This paper has been withdrawn by the authors because the results have already appeared elsewhere

openalex publication_date 2001/06/29 · arxiv created 2001/07/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study M(n,k,r), the number of orbits of (a₁,...,a_k)∈ Z_n^k | a₁+...+a_k = r (mod n) under the action of S_k. Equivalently, M(n,k,r) sums the partition numbers of an arithmetic sequence: M(n,k,r) = sum_{t ≥ 0} p(n-1,k,r+nt), where p(a,b,t) denotes the number of partitions of t into at most b parts, each of which is at most a. We derive closed formulas and various identities for such arithmetic partition sums. These results have already appeared in Elashvili/Jibladze/Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Alg. Combin. 10 (1999) 173-188, and the main result was also published by Von Sterneck in Sitzber. Akad. Wiss. Wien. Math. Naturw. Class. 111 (1902), 1567-1601 (see Lemma 2 and references in math.NT/9909121). Thanks to Don Zagier and Robin Chapman for bringing these references to our attention.

Related

• Number of partitions of modular integers (with an Appendix by P. Deligne)
• Partitions of ℤ_n into Arithmetic Progressions
• Partition Algebras and the Invariant Theory of the Symmetric Group
• Partition function of the cyclic group
• Rook sums in the symmetric group algebra