6-regular partitions: new combinatorial properties, congruences, and linear inequalities (Q6085332)

In this article, authors introduced infinite families of congruences modulo \(3\) (in arithmetic progression) for \(b_{6}(n)\) (i.e., \(6\)-regular partitions of \(n\)), and investigated its connections with \(Q_{2}(n)\) (i.e., the number of partitions on \(n\) into distinct parts not congruent to \(\pm2\) modulo \(2\)), also provided new combinatorial interpretations for \(b_{6}(n)\) and \(Q_{2}(n)\). Further, discovered new infinite families of linear inequalities involving Euler's partition function \(p(n)\). \textit{Q.-H. Hou} et al. [Adv. Appl. Math. 70, 32--44 (2015; Zbl 1327.05025)] discovered infinite infinite families of congruence relations module \(3, 5\) ans \(7\) for \(l\)-regular partitions with \(l\in\{3,5,6,7,10\}\), which is extended to other choices of primes and discussed in Theorem \(1.1\). Several properties of \(b_{6}(n)\) and \(Q_{2}(n)\) and their interrelationships are discussed in Theorems \(1.3, 1.5, 1.8, 1.10, 1.12\) and \(1.14\), and proofs of these theorems are based on generating functions. Further, gave combinatorial proofs for the Theorems \(1.3, 1.12(ii)\) and \(1.14(ii)\), also left an open problem for the combinatorial proofs of remaining theorems. Also, proposed conjectures for near infinite families of linear inequalities for the partitions functions \(b_{6}(n), Q_{2}(n)\) and \(p(n)\). All the results presented in this article are interesting and useful for further research work in this area.