Title

Fibonomial Tilings and Other Up-Down Tilings

Graduation Year

2016

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Science

Department

Mathematics

Reader 1

Arthur Benjamin

Reader 2

Nicholas Pippenger

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Rights Information

© 2016 Robert L Bennett

Abstract

The Fibonomial coefficients are a generalization of the binomial coefficients with a rather nice combinatorial interpretation. While the ordinary binomial coefficients count lattice paths in a grid, the Fibonomial coefficients count the number of ways to draw a lattice path in a grid and then Fibonacci-tile the regions above and below the path in a particular way. We may forgo a literal tiling interpretation and, instead of the Fibonacci numbers, use an arbitrary function to count the number of ways to "tile" the regions of the grid delineated by the lattice path. When the function is a combinatorial sequence such as the Lucas numbers or the q-numbers, the total number of "tilings" is some multiple of a generalized binomial coefficient corresponding to the sequence chosen.