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6-regular partitions: new combinatorial properties, congruences, and linear inequalities

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We consider the number of the 6-regular partitions of n, \(b_6(n)\), and give infinite families of congruences modulo 3 (in arithmetic progression) for \(b_6(n)\). We also consider the number of the partitions of n into distinct parts not congruent to \(\pm 2\) modulo 6, \(Q_2(n)\), and investigate connections between \(b_6(n)\) and \(Q_2(n)\) providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler’s partition function p(n). Infinite families of linear inequalities involving the 6-regular partition function \(b_6(n)\) and the distinct partition function \(Q_2(n)\) are proposed as open problems.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, MA, 01610, USA

    Cristina Ballantine

  2. Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering Research Center, University Politehnica of Bucharest, 060042, Bucharest, Romania

    Mircea Merca

  3. Academy of Romanian Scientists, 050044, Bucharest, Romania

    Mircea Merca

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  1. Cristina Ballantine
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  2. Mircea Merca
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Correspondence to Cristina Ballantine.

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Ballantine, C., Merca, M. 6-regular partitions: new combinatorial properties, congruences, and linear inequalities. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 159 (2023). https://doi.org/10.1007/s13398-023-01492-w

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