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Generalizations of Stanley’s Theorem: Combinatorial Proofs and Related Inequalities

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We give combinatorial proofs for the generalizations of Stanley’s Theorem given in Andrews and Merca (Math Stud 89(1–2): 175–180, 2020). These involve the total number \(b_k(n)\) of parts equal to k in all partitions of n. We also introduce several infinite families of linear inequalities involving \(b_k(n)\).

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We thank the anonymous referee for pointing out the refinement of Theorem 1 mentioned at the end of Sect. 2.1, as well as the simpler argument for the proof of Theorem 2 mentioned at the end of Sect. 2.2.

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Correspondence to Cristina Ballantine.

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Ballantine, C., Merca, M. Generalizations of Stanley’s Theorem: Combinatorial Proofs and Related Inequalities. Mediterr. J. Math. 19, 20 (2022).

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