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New combinatorial interpretations of the binomial coefficients
Paul M. Rakotomamonjy, Sandrataniaina R. Andriantsoa · 2019-03-15 · via math.CO updates on arXiv.org

Using generating functions and some trivial bijections, we show in this paper that the binomial coefficients count the set of (123,132) and (123,213)-avoiding permutations according to the number of crossings. We also define a q-tableau of power of two and prove that it counts the set of (213,312) and (132,312)-avoiding permutations according to the number of crossings.

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