Variations on a Generating-Function Theme: Enumerating Compositions with Parts Avoiding an Arithmetic Sequence

A \Def{composition} of a positive integer $n$ is a $k$-tuple $(ł_1, ł_2, \dots, ł_k) \in \Z_{> 0}^k$ such that $n = ł_1 + ł_2 + \dots + ł_k$. Our goal is to enumerate those compositions whose parts $ł_1, ł_2, \dots, ł_k$ avoid a fixed arithmetic sequence. When this sequence is given by the even integers (i.e., all parts of the compositions must be odd), it is well known that the number of compositions is given by the Fibonacci sequence. A much more recent theorem says that when the parts are required to avoid all multiples of a given integer $k$, the resulting compositions are counted by a sequence given by a Fibonacci-type recursion of depth $k$. We extend this result to arbitrary arithmetic sequences. Our main tool is a lemma on generating functions which is no secret among experts but deserves to be more widely known.

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