Heubach, Silvia; Mansour, Toufik Counting rises, levels, and drops in compositions. (English) Zbl 1077.05003 Integers 5, No. 1, Paper A11, 24 p. (2005). 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. Reviewer: Astrid Reifegerste (Hannover) Cited in 7 Documents MSC: 05A05 Permutations, words, matrices 05A15 Exact enumeration problems, generating functions Keywords:palindromic compositions; Carlitz compositions; partitions; generating functions PDFBibTeX XMLCite \textit{S. Heubach} and \textit{T. Mansour}, Integers 5, No. 1, Paper A11, 24 p. (2005; Zbl 1077.05003) Full Text: arXiv EuDML Online Encyclopedia of Integer Sequences: Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, .... Coefficients of expansion of (1-x)/(1-2*x) in powers of x. a(n) = 2^floor(n/2). a(n) = ((3*n+1)*2^n - (-1)^n)/9. a(n) = n*Fibonacci(n). Number of fixed points in all 231-avoiding involutions in S_n. Number of levels in the compositions of n with odd summands. Triangle T(n,m) read by rows: number of rises (drops) in the compositions of n with m parts, m>=2.