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Counting rises, levels, and drops in compositions. (English) Zbl 1077.05003

The authors derive the generating function for the number of compositions with parts in a given set with respect to the number of their parts, rises, levels, and drops. (The consecutive parts \(\sigma_i\) and \(\sigma_{i+1}\) of a composition \(\sigma\) are called a rise/level/drop if \(\sigma_i<\sigma_{i+1}\)/\(\sigma_i=\sigma_{i+1}\)/\(\sigma_i>\sigma_{i+1}\).) The result covers series of partial enumerations given previously by various authors.
Analogous results are also presented for compositions with additional properties, as palindromic compositions, Carlitz compositions and partitions.

MSC:

05A05 Permutations, words, matrices
05A15 Exact enumeration problems, generating functions
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