A unimodal sequence with mode at a quarter length

TL;DR

This paper proves that the sequence counting partitions with specific properties is strongly unimodal with a mode at a quarter of the largest hook length, using recurrence relations and zero distribution analysis.

Contribution

It establishes the unimodality of the partition sequence and analyzes the zero distribution of related generating functions using advanced algebraic methods.

Findings

Sequence $A(n,m)$ is strongly unimodal with mode at (n-1)/4 for n ≥ 6.

Zeros of generating functions lie on the left half of the circle |z-1|=2.

Zeros are densely distributed on a half circle, indicating specific root geometry.

Abstract

We show that the number $A(n,m)$ of partitions with $m$ even parts and largest hook length $n$ is strongly unimodal with mode [(n-1)/4] for $n\ge 6$. We establish this result by induction, using a $5$-term recurrence due to Lin, Xiong and Yan, and two $4$-term recurrences obtained by Zeilberger's algorithm. The sequence $A(n,m)$ is not log-concave. Using M\"obius transformation and the method of interlacing zeros, we obtain that every zero of every generating function $\sum_m A(n,m)z^m$ lies on the left half part of the circle |z-1|=2. Moreover, as a direct application of Wang and Zhang's characterization of root geometry of polynomial sequences that satisfy a recurrence of type $(1,1)$, we see that all these zeros are densely distributed on the half circle.