A combinatorial argument is used to explain the integrality of Fibonomial coefficients and their generalizations. The numerator of the Fibonomial coeffcient counts tilings of staggered lengths, which can be decomposed into a sum of integers, such that each integer is a multiple of the denominator of the Fibonomial coeffcient. By colorizing this argument, we can extend this result from Fibonacci numbers to arbitrary Lucas sequences.
© 2008/2009 The Fibonacci Association
Benjamin, A.T., & Plott, S.S. (2008/2009). A combinatorial approach to Fibonomial coefficients. Fibonacci Quarterly, 46/47(1): 7-9.
First published in the Fibonacci Quarterly, vol. 46/47, no. 1 (February 2008/2009), by the Fibonacci Association.
This article is also available at http://www.fq.math.ca/46_47-1.html.
Errata (attached as an additional file) published in the Fibonacci Quarterly, vol 48, no. 3 (August 2010), by the Fibonacci Association.