Abstract
ABSTRACT We show that the number of partitions with m even parts and largest hook length n is strongly unimodal with mode for . 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 is not log-concave. Using Möbius transformation and the method of interlacing zeros, we obtain that every zero of every generating polynomial lies on the left half part of the circle . Moreover, as an application of Wang and Zhang's characterization of root geometry of polynomial sequences that satisfy a recurrence, we confirm that the zeros are densely distributed on the half circle.
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