Abstract
The minimal excludant, or mex-function, on a set of positive integers is the smallest positive integer not in it. Andrews and Newman defined the mex-function \(\text{ mex}_{A,a}(\lambda )\) to be the smallest positive integer congruent to a modulo A that is not part of partition \(\lambda \), and denote by \(p_{A,a}(n)\) (reps. \(\overline{p}_{A,a}(n)\)) the number of partitions \(\lambda \) of n satisfying \(\text{ mex}_{A,a}(\lambda )\equiv a \text{(reps. } a+A{\text{) } }\pmod {2A},\) and found numerous surprising identities involving these functions. Motivated by the above results, in this paper, we prove that the number of the partitions of n with an even (resp. odd) number of even parts equals the mex-function \(p_{4,2}(n)\) (reps. \(\overline{p}_{4,2}(n)\)). We also derive several identities connecting the differences of two mex-functions with partitions restricted by certain congruences, which develops the work of Dhar, Mukhopadhyay, and Sarma. Furthermore, we extend the mex-function to overpartitions and study the relevant properties.
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References
Andrews, G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)
Andrews, G.E.: Concave compositions. Electron. J. Combin. 18, 6 (2011)
Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part I. Springer, New York (2005)
Andrews, G.E., Berndt, B.C.: Ramanujan’s Lost Notebook, Part II. Springer, New York (2009)
Andrews, G.E., Eriksson, K.: Integer Partitions. Cambridge University Press, Cambridge (2004)
Andrews, G.E., Newman, D.: Partitions and the minimal excludant. Ann. Comb. 23, 249–254 (2019)
Andrews, G.E., Newman, D.: The minimal excludant in integer partitions. J. Integer Seq. 23, 2–20 (2020)
Berndt, B.C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society, Providence (2006)
Berndt, B.C., Kim, B., Yee, A.J.: Ramanujan’s lost notebook: combinatorial proofs of identities associated with Heine’s transformation or partial theta functions. J. Combin. Theory Ser. A 117, 957–973 (2010)
Berndt, B.C., Yee, A.J.: Combinatorial proofs of identities in Ramanujan’s lost notebook associated with the Rogers-Fine identity and false theta functions. Ann. Comb. 7, 409–423 (2003)
Blecksmith, R., Brillhart, J., Gerst, I.: Parity results for certain partition functions and identities similar to theta function identities. Math. Comp. 48, 29–38 (1987)
Blecksmith, R., Brillhart, J., Gerst, I.: Some infinite product identities. Math. Comp. 51, 301–314 (1988)
Cooper, S., Hirschhorn, M.: On some infinite product identities. Rocky Mt. J. Math. 31, 131–139 (2001)
Corteel, S., Lovejoy, J.: Overpartitions. Trans. Am. Math. Soc. 356, 1623–1635 (2004)
Dabbagh, M.Z.: \(2\)-colored Rogers–Ramanujan partition identities. http://arxiv.org/abs/2203.15378
Dhar, A., Mukhopadhyay, A., Sarma, R.: New relations of the mex with other partition statistics. http://arxiv.org/abs/2201.05997v1
Fine, N.J.: Basic Hypergeometric Series and Applications. American Mathematical Society, Providence (1988)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Cambridge University Press, Cambridge (2004)
MacMahon, P.A.: Combinatory Analysis, vol. II. American Mathematical Society, Providence (2001)
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This work is supported by the National Natural Science Foundation of China (Grant No. 11901074), the General Program of Natural Science Foundation of Chongqing (Grant No. cstc2019jcyj-msxmX0104) and the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202200614).
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Yang, J.Y.X., Zhou, L. Identities on mex-related partitions. Ramanujan J 63, 157–181 (2024). https://doi.org/10.1007/s11139-023-00744-y
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DOI: https://doi.org/10.1007/s11139-023-00744-y