Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A reciprocity on finite abelian groups involving zero-sum sequences   In this paper, we present a reciprocity on finite abelian groups involving
zero-sum sequences. Let $G$ and $H$ be finite abelian groups with
$(|G|,|H|)=1$. For any positive integer $m$, let $\mathsf M(G,m)$ denote the
set of all zero-sum sequences over $G$ of length $m$. We have the following
reciprocity $$|\mathsf M(G,|H|)|=|\mathsf M(H,|G|)|.$$ Moreover, we provide a
combinatorial interpretation of the above reciprocity using ideas from rational
Catalan combinatorics. We also present and explain some other symmetric
relationships on finite abelian groups with methods from invariant theory.
Among others, we partially answer a question proposed by Panyushev in a
generalized version.

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The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg