Abstract
The minimal excludant, or “mex” function, on a set S of positive integers is the least positive integer not in S. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations of one of the families of functions Andrews and Newman studied, namely \(p_{t,t}(n)\), and provide complete parity characterizations of \(p_{1,1}(n)\) and \(p_{3,3}(n)\). In this article, we study the parity of \(p_{t,t}(n)\) when \(t=2^{\alpha }, 3\cdot 2^{\alpha }\) for all \(\alpha \ge 1\). We prove that \(p_{2^{\alpha },2^{\alpha }}(n)\) and \(p_{3\cdot 2^{\alpha }, 3\cdot 2^{\alpha }}(n)\) are almost always even for all \(\alpha \ge 1\). Using a result of Ono and Taguchi on nilpotency of Hecke operators, we also find infinite families of congruences modulo 2 satisfied by \(p_{2^{\alpha },2^{\alpha }}(n)\) and \(p_{3\cdot 2^{\alpha }, 3\cdot 2^{\alpha }}(n)\) for all \(\alpha \ge 1\).
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References
Allouche, J.-P., Goldmakher, L.: Mock characters and the Kronecker symbol. J. Number Theory 192, 356–372 (2018)
Andrews, G.E., Garvan, F.G.: Dysons crank of a partition. Bull. Am. Math. Soc. 18, 167–171 (1988)
Andrews, G.E., Newman, D.: The minimal excludant in integer partitions. J. Integer Sequences 23 (2020), Article 20.2.3
Aricheta, V.M.: Congruences for Andrews \((k, i)\)-singular overpartitions. Ramanujan J. 43, 535–549 (2017)
da Silva, R., Sellers, J. A.: Parity considerations for the Mex-related partition functions of Andrews and Newman. J. Integer Sequences 23 (2020), Article 20.5.7
Dyson, F.J.: Some guesses in the theory of partitions. Eureka 8, 10–15 (1944)
Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Springer, New York (1991)
Ono, K.: Parity of the partition function in arithmetic progressions. J. Reine Angew. Math. 472, 1–15 (1996)
Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and \(q\)-series, CBMS Regional Conference Series in Mathematics, 102. American Mathematical Society, Providence (2004)
Ono, K., Taguchi, Y.: 2-adic properties of certain modular forms and their applications to arithmetic functions. Int. J. Number Theory 1, 75–101 (2005)
Serre, J.-P.: Valeurs propres des opérateurs de Hecke modulo \(\ell \). Astérisque 24, 109–117 (1975)
Serre, J.-P.: Divisibilité de certaines fonctions arithmétiques. Séminaire Delange-Pisot-Poitou, Théorie des nombres 16, 1–28 (1974)
Serre, J.-P.: Divisibilité des coefficients des formes modulaires de poids entier. C. R. Acad. Sci. Paris (A) 279, 679–682 (1974)
Sloane N.J.A. et al.: The on-line encyclopedia of integer sequences (202). https://oeis.org
Tate, J.: \({\mathbb{Q}}\) unramified outside 2. In: Arithmetic geometry: conference on arithmetic geometry with an emphasis on Iwasawa theory, March 15–18, 1993, Arizona State University, vol. 174, No. 174. American Mathematical Society, Providence (1994)
Acknowledgements
We thank the referee for many valuable comments. We are extremely grateful to Ken Ono for previewing a preliminary version of this paper and for his helpful comments. We are indebted to Victor Manuel Aricheta for many fruitful discussions while preparing this article.
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Barman, R., Singh, A. On mex-related partition functions of Andrews and Newman. Res. number theory 7, 53 (2021). https://doi.org/10.1007/s40993-021-00284-8
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DOI: https://doi.org/10.1007/s40993-021-00284-8