We consider the number of k’s in all the partitions of n, and provide new connections between partitions and functions from multiplicative number theory. In this context, we obtain a new generalization of Euler’s pentagonal number theorem and provide combinatorial interpretations for some special cases of this general result.
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Andrews, G.E.: The Theory of Partitions. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1998)
Andrews, G.E., Merca, M.: The truncated pentagonal number theorem. J. Comb. Theory Ser. A 119, 1639–1643 (2012)
Andrews, G.E., Merca, M.: On the number of even parts in all partitions of \(n\) into distinct parts. Ann. Comb. 24, 47–54 (2020)
Berkovich, A., Garvan, F.G.: Some observations on Dyson’s new symmetries of partitions. J. Comb. Theory Ser. A 100, 61–93 (2002)
Christopher, D.: Partitions with fixed number of sizes. J. Integer Seq. 18, 15.11.5 (2015). https://www.emis.de/journals/JIS/VOL18/Christopher/chris7.pdf
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979)
Klazar, M.: Counting even and odd partitions. Am. Math. Mon. 110, 527–532 (2003)
Knopfmacher, A., Mays, M.E.: The sum of distinct parts in compositions and partitions. Bull. Inst. Comb. Appl. 25, 66–78 (1999)
Liu, J.-C.: Some finite generalizations of Euler’s pentagonal number theorem. Czechoslov. Math. J. 67, 525–531 (2017)
Merca, M.: A new look on the generating function for the number of divisors. J. Number Theory 149, 57–69 (2015)
Merca, M.: Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer. J. Number Theory 160, 60–75 (2016)
Merca, M.: New relations for the number of partitions with distinct even parts. J. Number Theory 176, 1–12 (2017)
Merca, M.: The Lambert series factorization theorem. Ramanujan J. 44(2), 417–435 (2017)
Merca, M.: New connections between functions from additive and multiplicative number theory. Mediterr. J. Math. 15, 36 (2018)
Merca, M., Schmidt, M.D.: A partition identity related to Stanley’s theorem. Am. Math. Mon. 125(10), 929–933 (2018)
Merca, M., Schmidt, M.D.: The partition function \(p(n)\) in terms of the classical Möbius function. Ramanujan J. 49, 87–96 (2019)
Merca, M., Schmidt, M.D.: Generating special arithmetic functions by Lambert series factorizations. Contrib. Discret. Math. 14(1), 31–45 (2019)
Merca, M., Schmidt, M.D.: Factorization theorems for generalized Lambert series and applications. Ramanujan J. 51, 391–419 (2020)
Schmidt, M.D.: New recurrence relations and matrix equations for arithmetic functions generated by Lambert series. Acta Arith. 181, 355–367 (2017)
Shanks, D.: A short proof of an identity of Euler. Proc. Am. Math. Soc. 2, 747–749 (1951)
Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences. Published electronically (2020). https://oeis.org
Tani, N.: Enumeration of the partitions of an integer into parts of a specified number of different sizes and especially two sizes. J. Integer Seq. 14, 11.3.6 (2011). https://cs.uwaterloo.ca/journals/JIS/VOL14/Tani/tani7.pdf
Warnaar, S.O.: \(q\)-hypergeometric proofs of polynomial analogues of the triple product identity, Lebesgue’s identity and Euler’s pentagonal number theorem. Ramanujan J. 8, 467–474 (2004)
Wilf, H.: Three problems in combinatorial asymptotics. J. Comb. Theory Ser. A 35, 199–207 (1983)
The author wish to thank the anonymous referees for a very careful reading of the original manuscript.
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Merca, M. Generalized Lambert Series and Euler’s Pentagonal Number Theorem. Mediterr. J. Math. 18, 29 (2021). https://doi.org/10.1007/s00009-020-01663-8
- pentagonal number theorem
- Lambert series
Mathematics Subject Classification