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Polynomization of the Bessenrodt–Ono Inequality
Annals of Combinatorics ( IF 0.6 ) Pub Date : 2020-08-25 , DOI: 10.1007/s00026-020-00509-0
Bernhard Heim , Markus Neuhauser , Robert Tröger

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 \(P_n(x)\). We prove for all real numbers \(x >2 \) and \(a,b \in \mathbb {N}\) with \(a+b >2\) the inequality:

$$\begin{aligned} P_a(x) \, \cdot \, P_b(x) > P_{a+b}(x). \end{aligned}$$

We show that \(P_n(x) < P_{n+1}(x)\) for \(x \ge 1\), 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: \(P_2(-3+ \sqrt{10}) = P_{3}(-3 + \sqrt{10})\).



中文翻译:

贝森罗特-奥诺不等式的多项式

在本文中,我们遵循吉安·卡洛·罗塔(Gian-Carlo Rota)的建议,以多项式根为基础研究组合论和数论问题,从而研究了贝森罗德-奥诺不等式的推广。我们将nk个彩色分区的数量视为多项式\(P_n(x)\)的特殊值。我们证明所有实数\(x> 2 \)\(a,b \ in \ mathbb {N} \)中的\(a + b> 2 \)不等式:

$$ \ begin {aligned} P_a(x)\,\ cdot \,P_b(x)> P_ {a + b}(x)。\ end {aligned} $$

我们表明,\(P_N(X)<P_ {N + 1}(x)的\)\(X \ GE 1 \) ,从而推广\(P(N)<P(N + 1)\) ,其中pn)表示分区函数。最后,对于较小的值,我们可以观察到相反的情况,因为,例如:\(P_2(-3+ \ sqrt {10})= P_ {3}(-3 + \ sqrt {10})\)

更新日期:2020-08-25
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