A new generalization of the minimal excludant arising from an analogue of Franklin’s identity. (English) Zbl 1509.05029
Summary: Euler’s classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. F. Franklin [J. Hopkins circ. II. 72 (1883; JFM 15.0131.04)] gave a generalization by considering partitions with exactly \(j\) different multiples of \(r\), for a positive integer \(r\). We prove an analogue of Franklin’s identity by studying the number of partitions with \(j\) multiples of \(r\) in total and in the process, discover a natural generalization of the minimal excludant (mex) which we call the \(r\)-chain mex. Further, we derive the generating function for \(\sigma_{r c} \mathrm{mex}(n)\), the sum of \(r\)-chain mex taken over all partitions of \(n\), thereby deducing a combinatorial identity for \(\sigma_{r c} \mathrm{mex}(n)\), which neatly generalizes the result of G. E. Andrews and D. Newman [Ann. Comb. 23, No. 2, 249–254 (2019; Zbl 1458.11011); J. Integer Seq. 23, No. 2, Article 20.2.3, 11 p. (2020; Zbl 1441.11265)] for \(\sigma \mathrm{mex}(n)\), the sum of mex over all partitions of \(n\).
MSC:
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A17 | Combinatorial aspects of partitions of integers |
| 11P81 | Elementary theory of partitions |
| 11P83 | Partitions; congruences and congruential restrictions |
Software:
OEISReferences:
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