PROOF OF A CONJECTURE OF CHAN, ROBBINS, AND YUEN

Doron ZEILBERGER ^{1}

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)/2dimensional polytope is given in terms of the product of the first n − 1 Catalan numbers.

Chan, Robbins, and Yuen[CRY] conjectured that the cardinality of a certain set of triangular arrays A_{n} defined in pp. 6-7 of [CRY] 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, CT_{z}f(z) is the coeff. of z^{0} in the formal Laurent expansion of f(z) (that always exists)):

CT_{x_{n}} … CT_{x_{1}} ∏_{i=1}^{n}(1 − x_{i})^{−2} ∏_{1≤i<j≤n}(x_{j} − x_{i})^{−1} = ∏_{i=1}^{n} \frac{1}{i + 1}\binom{2i}{i} .  (CRY)

But this is just the special case a = 2, b = 0, c = 1/2, of the Morris Identity[M] (where we made some trivial changes of discrete variables, and ‘shadowed’ it)

CT_{x_{n}} … CT_{x_{1}} ∏_{i=1}^{n}(1 − x_{i})^{−a} ∏_{i=1}^{n} x_{i}^{−b} ∏_{1≤i<j≤n}(x_{j} − x_{i})^{−2c} = \frac{1}{n!} ∏_{j=0}^{n−1} \frac{Γ(a + b + (n − 1 + j)c)Γ(c)}{Γ(a + jc)Γ(c + jc)Γ(b + jc + 1)} .

(Chip)

To show that the right side of (Chip) reduces to the right side of (CRY) upon the specialization a = 2, b = 0, c = 1/2, do the plugging in the former and call it M_{n}. Then manipulate the products to simplify M_{n}/M_{n−1}, and then use Legendre’s duplication formula Γ(z)Γ(z + 1/2) = Γ(2z)Γ(1/2)/2^{2z−1} three times, and voilà, up pops the Catalan number \binom{2n}{n}/(n + 1).

Remarks: 1. By converting the left side of (Chip) into a contour integral, we get the same integrand as in the Selberg integral (with a → −a, b → −b − 1, c → −c). Aomoto’s proof of the Selberg integral (SIAM J. Math. Anal. 18(1987), 545-549) goes verbatim. 2. Conjecture 2 in [CRY] follows in the same way, from (the obvious contour-integral analog of) Aomoto’s extension of Selberg’s integral. Introduce a new variable t, stick CT_{t}t^{−k} in front of (CRY), and replace (1 − x_{i})^{−2} by (1 − x_{i})^{−1}(t + x_{i}/(1 − x_{i})). 3. Conjecture 3 follows in the same way from another specialization of (Chip).

References

[CRY] Clara S. Chan, David P. Robbins, and David S. Yuen, On the volume of a certain polytope,

math.CO/9810154.

[M] Walter Morris, “Constant term identities for finite and affine root systems, conjectures and theorems”, Ph.D. thesis, University of Wisconsin, Madison, Wisconsin, 1982.

1 Department of Mathematics, Temple University, Philadelphia, PA 19122, USA. zeilberg@math.temple.edu

http://www.math.temple.edu/~zeilberg/ . Nov. 17, 1998. Supported in part by the NSF.