CONGRUENCES FOR BIPARTITIONS WITH ODD DESIGNATED SUMMANDS

M.S.Mahadeva Naika; Harishkumar T; M Prasad; T.N Veeranayaka

doi:10.56827/SEAJMMS.2023.1902.1

M.S.Mahadeva Naika Department of Mathematics, Bengaluru City University, Central College Campus, Bengaluru - 560001, Karnataka, INDIA
Harishkumar T Department of Mathematics, Bengaluru City University, Central College Campus, Bengaluru - 560001, Karnataka, INDIA
M Prasad Department of Mathematics, PES College of Engineering, Mandya - 571401, Karnataka, INDIA
T.N Veeranayaka Department of Mathematics, Bengaluru City University, Central College Campus, Bengaluru - 560001, Karnataka, INDIA

DOI: https://doi.org/10.56827/SEAJMMS.2023.1902.1

Keywords: Designated summands, Congruences, Theta functions, Dissections.

Abstract

Andrews, Lewis and Lovejoy investigated a new class of partitions with designated summands by taking ordinary partitions and tagging exactly one of each part size. Let B2(n) count the number of bipartitions of n with designated summands in which all parts are odd. In this work, we establish many infinite families of congruences modulo powers of 2 and 3 for B2(n). For example, for each n > 0 and Alpha > 0, B2 ???? 48 · 52Alpha+2n + a1 · 52Alpha+1 ≡ 0 (mod 9), where a1 ∈ {88, 136, 184, 232}.