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Combinations with successions and Fibonacci numbers. (English)
Fibonacci Q. 45, No. 2, 104-114 (2007).
From the author’s abstract and introduction: “In his book [{\it J. Riordan}, Combinatorial identities. (New York-London-Sydney): John Wiley and Sons, Inc. (1968; Zbl 0194.00502)] John Riordan considers the enumeration of $k$-combinations of $\{1,2,\dots,n\}$ which contain a specified number of pairs of consecutive integers or successions. We study variations and generalizations of the original idea of combinations with successions, and obtain enumerative recurrences and formulas. We show that formulas for combinations with successions also enumerate a class of restricted compositions and the multi-step Fibonacci numbers. ... The exposition mostly resembles that of a fundamental paper by the author [{\it A. O. Munagi}, "Set partitions with successions and separations." Int. J. Math. Math. Sci. 2005, No. 3, 451-463 (2005; Zbl 1076.05009)], which deals with set partitions."
Reviewer: William G. Brown (Montréal)