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Congruence successions in compositions

Abstract : A composition is a sequence of positive integers, called parts, having a fixed sum. By an m-congruence succession, we will mean a pair of adjacent parts x and y within a composition such that x=y(modm). Here, we consider the problem of counting the compositions of size n according to the number of m-congruence successions, extending recent results concerning successions on subsets and permutations. A general formula is obtained, which reduces in the limiting case to the known generating function formula for the number of Carlitz compositions. Special attention is paid to the case m=2, where further enumerative results may be obtained by means of combinatorial arguments. Finally, an asymptotic estimate is provided for the number of compositions of size n having no m-congruence successions.
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Toufik Mansour, Mark Shattuck, Mark C Wilson. Congruence successions in compositions. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2014, Vol. 16 no. 1 (1), pp.327--338. ⟨10.46298/dmtcs.1252⟩. ⟨hal-01179220⟩



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