Some Elementary Congruences for the Number of Weighted Integer Compositions
Computer Science Department
Goethe University Frankfurt am Main
60325 Frankfurt am Main
An integer composition of a nonnegative integer n
is a tuple
of nonnegative integers whose sum is n
called the parts
composition. For fixed number k
of parts, the number of
integer compositions (also called f-colored integer
the literature), in which each part size s
may occur in f
) different colors, is given by the extended binomial coefficient
We derive several congruence properties for
most of which are analogous to those for ordinary
binomial coefficients. Among them is the parity of
Babbage's congruence, Lucas' theorem, etc. We also give congruences
), 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.
Full version: pdf,
(Concerned with sequences
Received September 15 2014;
revised version received February 20 2015.
Published in Journal of Integer Sequences, March 25 2015.
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