Extensions of Generalized Binomial Coefficients

Richard L. Ollerton and Anthony G. Shannon

A chapter in Applications of Fibonacci Numbers, 2004, pp 187-199 from Springer

Abstract: Abstract Bondarenko gives an excellent account of the history and properties of the generalized binomial coefficient [2] . He describes this coefficient, written % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa % qabeGabaaabaGaamOBaaqaaiaad2gaaaaacaGLOaGaayzkaaWaaSba % aSqaaiaadofaaeqaaaaa!3A71! $${\left( {\begin{array}{*{20}{c}} n \\ m \\ \end{array}} \right)_S}$$ for n, m ≥ 0 and s ≥ 1, as the number of ways m objects can be placed in n cells, each of which holds a maximum of s - 1 objects. The ordinary binomial coefficients are obtained when s = 2. This description tacitly assumes that empty cells produce distinct arrangements, that the objects are placed in the cells in a given order and that the order of the objects within the cells is not important.

Keywords: 11B39; 11B65; 15A36 (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-0-306-48517-6_19

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