Fractional partitions and conjectures of Chern-Fu-Tang and\n Heim-Neuhauser

2020/11/17 by Kathrin Bringmann, Bringmann, Kathrin, Ben Kane +5 · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2011.08874

openalex publication_date 2020/11/17 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Many papers have studied inequalities for partition functions. Recently, a\nnumber of papers have considered mixtures between additive and multiplicative\nbehavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser\ngave conjectures on inequalities for coefficients of powers of the generating\npartition function. These conjectures were posed in the context of colored\npartitions and the Nekrasov-Okounkov formula. Here, we study the precise size\nof differences of products of two such coefficients. This allows us to prove\nthe Chern-Fu-Tang conjecture and to show the Heim-Neuhauser conjecture in a\ncertain range. The explicit error terms provided will also be useful in the\nfuture study of partition inequalities. These are laid out in a user-friendly\nway for the researcher in combinatorics interested in such analytic questions.\n

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