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Robert Bennett

Picture of Robert Bennett.


Fibonomial Tilings and Other Up-Down Tilings

Arthur T. Benjamin
Second Reader(s)
Nicholas J. Pippenger


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 tile the regions above and below the path with squares and dominoes.

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.

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