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On the Number of Compositions of ***K*_{m} ×
*P*_{n}

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Liam Buttitta

**Abstract:**

In a paper written in 2001, Knopfmacher and Mays introduced the concept
of graph compositions. In another paper written in 2004, Ridley and Mays
proved a theorem on the number of compositions of the Cartesian product
of the path graph with another graph. We build on this theorem to create
an algorithm that calculates the closed-form solutions for the sequences
given by the numbers of compositions of the Cartesian product of the
path graph with complete graphs, and we apply this algorithm to produce
two new sequences in the *On-Line Encyclopedia of Integer Sequences*. We
also demonstrate some unexpected results that arise from these sequences,
and try to explain why they might occur.

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(Concerned with sequences
A000079
A000110
A078469
A344638
A346273.)

Received October 21 2021; revised versions received January 2 2022; February 1 2022.
Published in *Journal of Integer Sequences*,
March 27 2022.

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