Pascal's triangle is a very important structure in combinatorics: its entries, the binomial coefficients, answer a number of counting-related questions. I will define a set of functions, the Planar Rook monoid, whose structure is tied to Pascal's Triangle. Most of the connections between the monoid and Pascal's triangle are seen when we allow the functions to work as linear transformations on different vector spaces; I will show several examples which lead to algebraic proofs of famous binomial identities. These proofs give insight into the deep connection between the monoid and Pascal's Triangle.
Herbig, Kathryn E., "The Planar Rook Monoid" (2006). Mathematics, Statistics, and Computer Science Honors Projects. 1.
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