Proof of a combinatorial conjecture posed in "The Blimpy Shape of Heady-s and Taily-s Bit Strings"

TL;DR

This paper proves three conjectured properties of a function related to the blimpy shape of bit strings, including convexity, a formula for its negative maximum, and positive expectation, using new inequalities and monotonicity assumptions.

Contribution

It introduces and proves three properties of a specific function related to bit string analysis, confirming conjectures and providing new inequalities.

Findings

Confirmed discrete convexity of the function

Derived a formula for the maximum argument where the function is negative

Established positive expectation under certain probability models

Abstract

We demonstrate three properties conjectured to hold for a certain function by Levin (2025) in a study of the blimpy graphical shape of the number of bit strings with a given score under an interesting scoring system. The properties include discrete convexity, a simple formula for the greatest argument at which the function is negative, and a positive expectation under a certain probability function. A new set of inequalities which imply the latter is presented and proved under some monotonicity assumptions.