An upper bound on binomial coefficients in the de Moivre-Laplace form

TL;DR

This paper proposes a universal upper bound on binomial coefficients inspired by the de Moivre-Laplace approximation, with applications in Boolean function analysis, spectral dependencies, and sum-of-squares representations.

Contribution

It introduces a new upper bound on binomial coefficients valid across all parameters, extending the de Moivre-Laplace form and enabling various combinatorial and spectral analyses.

Findings

Derived a universal upper bound on binomial coefficients.

Applied the bound to estimate Boolean function continuations to bent functions.

Investigated spectral dependencies and sum-of-squares representations.

Abstract

We suggest an upper bound on binomial coefficients that holds over the entire parameter range and whose form repeats the form of the de Moivre-Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the number of continuations of a given Boolean function to bent functions, investigate dependencies into the Walsh-Hadamard spectra, obtain restrictions on the number of representations as sum of squares of integers bounded in magnitude.