2019/10/23 by Bernhard Heim, Heim, Bernhard, Markus Neuhauser +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT
paper · pdf · doi:10.48550/arxiv.1910.10413
arxiv created 2019/10/23 · arxiv updated 2019/10/24
In this paper we investigate the 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 Pn(x). We prove for all real numbers x >2 and a,b ∈ ℕ with a+b >2 the inequality Pa(x) ⋅ Pb(x) > Pa+b(x). We show that Pn(x) < Pn+1(x) for x ≥ 1, which generalizes 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: P2(-3+ √(10)) = P3(-3 + √(10)).