Some congruences modulo 2, 8 and 12 for Andrews' singular overpartitions

doi:10.1285/i15900932v38n1p101

Some congruences modulo 2, 8 and 12 for Andrews' singular overpartitions

Utpal Pore, S. N. Fathima

Abstract

Recently, G. E. Andrews defined combinatorial objects which he called $(k,i)$ -singular overpartitions, overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv\pm i\pmod k$ may be overlined. Let the number of $(k,i)$ -singular overpartitions of $n$ be\linebreak denoted by $\overline{C}_{k,i}(n)$ . Andrews and Chen, Hirschhorn and Sellers noted numerous congruences modulo $2$ for $\overline{C}_{3,1}(n)$ . The object of this paper is to obtain new congruences modulo $2$ for $\overline{C}_{20,5}(n)$ and modulo $8$ and $12$ for $\overline{C}_{3,1}(n)$ .

DOI Code: 10.1285/i15900932v38n1p101

Keywords: singular overpartition; congruence; generating function; sums of squares

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Some congruences modulo 2, 8 and 12 for Andrews' singular overpartitions

Abstract