Some Elementary Congruences for the Number of Weighted Integer Compositions

TL;DR

This paper explores various congruence properties of extended binomial coefficients related to weighted integer compositions, extending classical binomial coefficient results to a more general combinatorial setting.

Contribution

It introduces new congruence relations for extended binomial coefficients and weighted compositions, generalizing classical binomial coefficient properties like Lucas' theorem.

Findings

Derived congruences for extended binomial coefficients

Established parity and Babbage's congruence for weighted compositions

Provided prime criteria based on congruences for weighted compositions

Abstract

An integer composition of a nonnegative integer $n$ is a tuple $(\pi_1,\ldots,\pi_k)$ of nonnegative integers whose sum is $n$; the $\pi_i$'s are called the parts of the composition. For fixed number $k$ of parts, the number of $f$-weighted integer compositions (also called $f$-colored integer compositions in the literature), in which each part size $s$ may occur in $f(s)$ different colors, is given by the extended binomial coefficient $\binom{k}{n}_{f}$. We derive several congruence properties for $\binom{k}{n}_{f}$, most of which are analogous to those for ordinary binomial coefficients. Among them is the parity of $\binom{k}{n}_{f}$, Babbage's congruence, Lucas' theorem, etc. We also give congruences for $c_{f}(n)$, the number of $f$-weighted integer compositions with arbitrarily many parts, and for extended binomial coefficient sums. We close with an application of our results to prime criteria for weighted integer compositions.