From a Rogers’s identity to overpartitions

In this paper, the author provides an efficient linear recurrence relation for the number of partitions of n into parts not congruent to 0, \(\pm 1\), \(\pm 8\), \(\pm 9\) and \(10 \pmod {20}\). A simple criterion for deciding whether this number is odd or even is given as a corollary of this result. Some results involving overpartitions and partitions into distinct parts have been derived in this context.     

Congruences modulo 16, 32, and 64 for Andrews’s singular overpartitions

In a recent work, Andrews gave a definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be enumerated by the function \(\overline{C}_{k,i}(n) \) which denotes the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i \ (\mathrm{mod}\ k)\) may be overlined. Andrews, Chen, Hirschhorn and Sellers, and Ahmed and Baruah discovered numerous congruences modulo 2, 3, 4, 8, and 9 for \(\overline{C}_{3,1}(n)\). In this paper, we prove a number of congruences modulo 16, 32, and 64 for \(\overline{C}_{3,1}(n)\).     

Congruences for Andrews’ (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">i</Emphasis>)-singular overpartitions

Andrews recently defined new combinatorial objects which he called (k, i)-singular overpartitions and proved that they are enumerated by \(\overline{C}_{k,i}(n)\) which is the number of overpartitions of n in which no part is divisible by k and only the parts \(\equiv \pm i \pmod {k}\) may be overlined. Andrews further showed that \(\overline{C}_{3,1}(n)\) satisfies some Ramanujan-type congruences modulo 3. In this paper, we show that for any pair (k, i), \(\overline{C}_{k,i}(n)\) satisfies infinitely many Ramanujan-type congruences modulo any power of prime coprime to 6k. We also show that for an infinite family of k, the value \(\overline{C}_{3k,k}(n)\) is almost always even. Finally, we investigate the parity of \(\overline{C}_{4k,k}\).