No 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 > 2 and $$a,b \in \mathbb {N}$$ a , b ∈ N with $$a+b >2$$ a + b > 2 the inequality: $$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$ P a ( x ) · P b ( x ) > P a + b ( x ) . We show that $$P_n(x) < P_{n+1}(x)$$ P n ( x ) < P n + 1 ( x ) for $$x \ge 1$$ x ≥ 1 , which generalizes $$p(n) < p(n+1)$$ p ( n ) < 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 ) .

Inequalities are important features in the context of sequences of numbers and polynomials. The Bessenrodt-Ono inequality for partition numbers and Nekrasov-Okounkov polynomials has only recently been discovered. In this paper we study the log-concavity (Turán inequality) and Bessenrodt-Ono inequality for plane partitions and their polynomization.

Inequalities are important features in the context of sequences of numbers and polynomials. The Bessenrodt–Ono inequality for partition numbers and Nekrasov–Okounkov polynomials has only recently been discovered. In this paper we study the log-concavity (Turán inequality) and Bessenrodt–Ono inequality for plane partitions and their polynomization.