Log concavity for unimodal sequences

Area of Science:

• Combinatorics
• Number Theory
• Analytic Number Theory

Background:

• Log-concavity and higher Turán inequalities are significant in the study of partitions and modular forms.
• Existing analytic proofs for these properties often rely on precise asymptotic series with error terms.
• Recent advancements in false theta functions have yielded exact formulas for coefficients of mixed mock/false modular objects.

Purpose of the Study:

• To prove that the number of unimodal sequences of size n is log-concave.
• To utilize an exact formula for unimodal sequences to perform this calculation.
• To explore the applicability of the developed method to other related mathematical objects.

Main Methods:

• Derivation of an exact formula for unimodal sequences, building on recent work on false theta functions.
• Application of analytic techniques to the exact formula to establish log-concavity.
• Leveraging properties of mixed false modular forms.

Main Results:

• The number of unimodal sequences of size n is demonstrated to be log-concave.
• The study provides an analytic proof for the log-concavity of these combinatorial sequences.
• An exact formula for unimodal sequences, related to mixed false modular forms, is central to the proof.

Conclusions:

• The log-concavity of unimodal sequences is established.
• The methodology employed is expected to be applicable to other coefficients of mixed mock/false modular objects.
• This work contributes to the understanding of combinatorial sequences and their connection to modular forms.