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Congruence successions in compositions
A \emph{composition} is a sequence of positive integers, called \emph{parts},
having a fixed sum. By an \emph{-congruence succession}, we will mean a pair
of adjacent parts and within a composition such that . Here, we consider the problem of counting the compositions of
size according to the number of -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 , 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 having no -congruence
successions
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