ON A COMBINATORIAL INTERPRETATION OF THE BISECTIONAL PENTAGONAL NUMBER THEOREM

In this paper, we invoke the bisectional pentagonal number theorem to prove that the number of overpartitions of the positive integer n into odd parts is equal to twice the number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20 or 30 mod 32. This result allows us to experimentally discover new infinite families of linear partition inequalities involving Euler’s partition function p(n). In this context, we conjecture that for k > 0, the theta series 

has non-negative coefficients.