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arXiv · 1504.00389

Some Elementary Congruences for the Number of Weighted Integer Compositions

Steffen Eger
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Abstract

An integer composition of a nonnegative integer $n$ is a tuple $(π_1,\ldots,π_k)$ of nonnegative integers whose sum is $n$; the $π_i$'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.

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Steffen Eger. 2015-03-22. Some Elementary Congruences for the Number of Weighted Integer Compositions. https://arxiv.org/abs/1504.00389

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