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2016•Bulletin of the Australian Mathematical SocietyOpen access

THE 7-REGULAR AND 13-REGULAR PARTITION FUNCTIONS MODULO 3

Eric Boll, David Penniston

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Abstract

Let $b_{\ell }(n)$ denote the number of $\ell$ -regular partitions of $n$ . In this paper we establish a formula for $b_{13}(3n+1)$ modulo $3$ and use this to find exact criteria for the $3$ -divisibility of $b_{13}(3n+1)$ and $b_{13}(3n)$ . We also give analogous criteria for $b_{7}(3n)$ and $b_{7}(3n+2)$ .

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Let $b_{\ell }(n)$ denote the number of $\ell$ -regular partitions of $n$ . In this paper we establish a formula for $b_{13}(3n+1)$ modulo $3$ and use this to find exact criteria for the $3$ -divisibility of $b_{13}(3n+1)$ and $b_{13}(3n)$ . We also give analogous criteria for $b_{7}(3n)$ and $b_{7}(3n+2)$ .

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Available abstract

Let $b_{\ell }(n)$ denote the number of $\ell$ -regular partitions of $n$ . In this paper we establish a formula for $b_{13}(3n+1)$ modulo $3$ and use this to find exact criteria for the $3$ -divisibility of $b_{13}(3n+1)$ and $b_{13}(3n)$ . We also give analogous criteria for $b_{7}(3n)$ and $b_{7}(3n+2)$ .

Key concepts: Mathematics, Modulo, Partition (number theory), Combinatorics, Divisibility rule, Discrete mathematics

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