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Recurrence Identities of b-ary Partitions

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 220)

Abstract

Solving the b-ary partition problem, counting the number \(p_b(n)\) of partitions of n into powers of b, is a pursuit which dates back to Euler. The function \(p_b(n)\) satisfies a recurrence, and this note examines a family of identities which can be deduced by iterating the recurrence in a suitable way. These identities can then be used to calculate \(p_b(n)\) for large values of n. Further, these identities correspond to generating function identities involving a sequence of polynomials which have suggestive connections to Eulerian polynomials.

Keywords

  • Integer partitions
  • Partition functions
  • Recurrence
  • Congruences
  • Generating functions
  • Eulerian polynomials

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Notes

  1. 1.

    See Table 1 for values of \(p_b(bn)\) for small values of b and n. The expression \(p_b(bn)\) is chosen because by Theorem 3.1 the value of \(p_b(n)\) is constant on runs of b.

  2. 2.

    Note that the Stirling numbers \({{n}\brack {k}}\) and \({{n}\brace {k}}\) are defined on page 56.

  3. 3.

    Note that the Eulerian numbers \({\left\langle {n \atop k}\right\rangle }\) are defined on page 56.

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Acknowledgements

This research was supported, in part, under National Science Foundation Grants CNS-0958379, CNS-0855217, ACI-1126113 and the City University of New York High Performance Computing Center at the College of Staten Island.

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Blair, D. (2017). Recurrence Identities of b-ary Partitions. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_5

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