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Marking and Shifting a Part in Partitions

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Abstract

Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers–Ramanujan identities, the Göllnitz–Gordon identities, Euler’s odd = distinct theorem, and the Andrews–Gordon identities. Generalizations of each of these theorems are given where a single part is “marked” or weighted. This allows a single part to be replaced by a new larger part, “shifting” a part, and analogous combinatorial results are given in each case. Versions are also given for marking a sum of parts.

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References

  1. Andrews, G.E.: A generalization of the Göllnitz–Gordon partition theorems. Proc. Amer. Math. Soc. 18, 945–952 (1967)

    MathSciNet  MATH  Google Scholar 

  2. Andrews, G.E.: Partitions and Durfee dissection. Amer. J. Math. 101(3), 735–742 (1979)

    Article  MathSciNet  Google Scholar 

  3. Berkovich, A., Garvan, F.G.: Dissecting the Stanley partition function. J. Combin. Theory Ser. A 112(2), 277–291 (2005)

    Article  MathSciNet  Google Scholar 

  4. Berkovich, A., Grizzell, K.: A partition inequality involving products of two \(q\)-Pochhammer symbols. In: Alladi, K., Garvan, F., Yee, A.J. (eds.) Ramanujan 125, Contemp. Math., 627, pp. 25–39. Amer. Math. Soc., Providence, RI (2014)

  5. Göllnitz, H.: Partitionen mit Differenzenbedingungen. J. Reine Angew. Math. 225, 154–190 (1967)

    MathSciNet  MATH  Google Scholar 

  6. Gordon, B.: Some continued fractions of the Rogers–Ramanujan type. Duke Math. J. 32, 741–748 (1965)

    Article  MathSciNet  Google Scholar 

  7. Greene, J.: Bijections related to statistics on words. Discrete Math. 68(1), 15–29 (1988)

    Article  MathSciNet  Google Scholar 

  8. Kadell, K.W.J.: An injection for the Ehrenpreis Rogers–Ramanujan problem. J. Combin. Theory Ser. A 86(2), 390–394 (1999)

    Article  MathSciNet  Google Scholar 

  9. O’Hara, K., Stanton, D.: Refinements of the Rogers–Ramanujan identities. Exp. Math. 24(4), 410–418 (2015)

    Article  MathSciNet  Google Scholar 

  10. Stanton, D.: \(q\)-Analogues of Euler’s odd = distinct theorem. Ramanujan J. 19(1), 107–113 (2009)

    Article  MathSciNet  Google Scholar 

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  1. 4356 Preston Forest Drive, Blacksburg, VA, 24060, USA

    Kathleen O’Hara

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Dennis Stanton

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  1. Kathleen O’Hara
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  2. Dennis Stanton
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Correspondence to Dennis Stanton.

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O’Hara, K., Stanton, D. Marking and Shifting a Part in Partitions. Ann. Comb. 23, 935–951 (2019). https://doi.org/10.1007/s00026-019-00448-5

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  • DOI: https://doi.org/10.1007/s00026-019-00448-5

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