A unimodal sequence with mode at a quarter length

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 l...

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Bibliographic Details
Published inJournal of difference equations and applications Vol. 29; no. 7; pp. 763 - 778
Main Authors Liu, Max Y. C., Wang, David G. L.
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 03.07.2023
Taylor & Francis Ltd
Subjects
Algorithms
Geometric transformation
Log-concavity
Polynomials
real-rootedness
unimodality
Zeilberger's algorithm
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Summary: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.
ISSN:1023-6198
1563-5120
DOI:10.1080/10236198.2023.2253330