On unimodality problems in Pascal's triangle

TL;DR

This paper investigates the unimodality of specific binomial coefficient sequences in Pascal's triangle, confirming a conjecture and proposing new ones, with implications for understanding binomial coefficient properties.

Contribution

It proves a conjecture that certain binomial coefficient sequences are unimodal and introduces two broader conjectures on the topic.

Findings

Confirmed the unimodality conjecture for sequences in Pascal's triangle

Proposed two new conjectures on binomial coefficient unimodality

Provided theoretical insights into binomial coefficient sequence properties

Abstract

Many sequences of binomial coefficients share various unimodality properties. In this paper we consider the unimodality problem of a sequence of binomial coefficients located in a ray or a transversal of the Pascal triangle. Our results give in particular an affirmative answer to a conjecture of Belbachir et al which asserts that such a sequence of binomial coefficients must be unimodal. We also propose two more general conjectures.