| PROOF OF A CONJECTURE OF CHAN, ROBBINS, AND YUEN ∗ (2009) | |||||||||||||
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| Abstract. Using the celebrated Morris Constant Term Identity, we deduce a recent conjecture of Chan, Robbins, and Yuen (math.CO/9810154), that asserts that the volume of a certain n(n − 1)/2-dimensional polytope is given in terms of the product of the first n − 1 Catalan numbers. Key words. combinatorics, Catalan numbers, polytope. AMS subject classifications. 05-XX, 52B05. 1. Main Result. Chan, Robbins, and Yuen [1] conjectured that the cardinality of a certain set of triangular arrays An defined in pp. 6-7 of [1] equals the product of the first n − 1 Catalan numbers. It is easy to see that their conjecture is equivalent to the following constant term identity (for any rational function f(z) of a variable z, CTzf(z) is the coeff. of z 0 in the formal Laurent expansion of f(z) (that always exists)): (1.1) CTxn...CTx1 n∏ (1 − xi) i=1 | |||||||||||||
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