Polynomization of the Bessenrodt–Ono Inequality

Source abstract

Abstract In this paper, we investigate a generalization of the Bessenrodt–Ono inequality by following Gian–Carlo Rota’s advice in studying problems in combinatorics and number theory in terms of roots of polynomials. We consider the number of k -colored partitions of n as special values of polynomials $$P_n(x)$$ P n ( x ) . We prove for all real numbers $$x >2$$ x &gt; 2 and $$a,b \in \mathbb {N}$$ a , b ∈ N with $$a+b >2$$ a + b &gt; 2 the inequality: $$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$ P a ( x ) · P b ( x ) &gt; P a + b ( x ) . We show that $$P_n(x) < P_{n+1}(x)$$ P n ( x ) &lt; P n + 1 ( x ) for $$x \ge 1$$ x ≥ 1 , which generalizes $$p(n) < p(n+1)$$ p ( n ) &lt; p ( n + 1 ) , where p ( n ) denotes the partition function. Finally, we observe for small values, the opposite can be true, since, for example: $$P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})$$ P 2 ( - 3 + 10 ) = P 3 ( - 3 + 10 ) .