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Composition-theoretic series in partition theory

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Abstract

We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are k-gonal numbers; our proofs employ Ramanujan’s theta functions. We explore applications to lacunary q-series, and to a new class of composition-theoretic Dirichlet series.

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Notes

  1. We note a more “natural” formula with fewer parameters is not necessarily more convenient for applications.

  2. See [6, 23, 29] for further reading about this additive-multiplicative theory.

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Acknowledgements

The authors are grateful to George Andrews, Maurice Hendon, Mike Hirschhorn, Matthew Just, William Keith, Jeremy Lovejoy, Ken Ono, Cécile Piret, and James Sellers for comments that benefited our work. In particular, we thank C. Piret for advice on proving convergence in Proposition 1. Also, we thank Jonathan Bradley-Thrush for bringing to our attention the work of S. Mangeot.

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  1. Robert Schneider and Andrew V. Sills have contributed equally to this work.

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  1. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Robert Schneider

  2. Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, 30458, USA

    Andrew V. Sills

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Schneider, R., Sills, A.V. Composition-theoretic series in partition theory. Ramanujan J (2023). https://doi.org/10.1007/s11139-023-00780-8

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