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Séminaire Lotharingien de Combinatoire, B70h (2014), 24 pp.

# C. P. Anilkumar and Amritanshu Prasad

# Orbits of Pairs in Abelian Groups

**Abstract.**
We compute the number of orbits of pairs in a finitely generated torsion module (more generally, a module of bounded order) over a discrete valuation ring.
The answer is found to be a polynomial in the cardinality of the residue field whose coefficients are integers which depend only on the elementary divisors of the module, and not on the ring in question.
The coefficients of these polynomials are conjectured to be non-negative integers.

Received: May 29, 2013.
Accepted: June 20, 2014.
Final Version: June 26, 2014.

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