Inequalities for k-regular partitions

Research in Number Theory, vol. 12

Abstract

Abstract We build upon the work by Bessenrodt and Ono, as well as Beckwith and Bessenrodt concerning the combined additive and multiplicative behavior of the k -regular partition functions $$p_k(n)$$ p k ( n ) . Our focus is on addressing the solutions of the Bessenrodt–Ono inequality $$\begin{aligned} p_k(a) \, p_k(b) > p_k(a+b). \end{aligned}$$ p k ( a ) p k ( b ) > p k ( a + b ) . We determine the sets $$E_k$$ E k and $$F_k$$ F k consisting of all pairs ( a , b ), where we have equality or the opposite inequality. Bessenrodt and Ono previously determined the exception sets $$E_{\infty }$$ E ∞ and $$F_{\infty }$$ F ∞ for the partition function p ( n ). We prove by induction that $$E_k=E_{\infty }$$ E k = E ∞ and $$F_k=F_{\infty }$$ F k = F ∞ if and only if $$k \ge 10$$ k ≥ 10 . Beckwith and Bessenrodt used analytic methods to consider $$2 \le k \le 6$$ 2 ≤ k ≤ 6 , while Alanazi, Gagola, and Munagi studied the case $$k=2$$ k =

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