On Two Combinatorial Inequalities That Explain the Blimpy Shape of Heady-s and Taily-s Bit Strings

Abstract

We prove two inequalities introduced in our prior study of the graphical shape of the number of bit strings with a given score under an interesting scoring system. Generating functions are used to establish the inequalities, which in turn imply two of the salient graphical features, uni-modality near the zero score and shape asymmetry for positive versus negative scores. One inequality provides a lower bound on the expected value of a discrete random variable with probabilities proportional to a product of two binomial coefficients. The other inequality states that the expected value with respect to near central binomial coefficients of other binomial coefficients lying on an oblique ray in Pascal's triangle exceeds the expected value along an adjacent parallel ray to its left.

Disclosure

“OWLEDGMENT The author would like to thank Cheng-Shiun Leu and Jacob Fink for helpful discussions. DISCLOSURE STATEMENT The author has no conflicts of interest to report. Gemini 3 Flash was used to look up known generating functions and Claude AI was used to cross-check my asymptotic calculations. APPENDIX A We derive formulas (2.6) and (2.7) using the transfer theorems of singularity analysis (Flajolet and Sedgewick, 2009) for generating functions of the form Gα , β ( x) = F”

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Classification

Computational experiments or data processing

Multiplier

3

Verified

2026-08-24T00:34:15.228Z

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