The combinatorics of a three-line circulant determinant

We study the polynomial , where ω is a primitivepth root of unity. This polynomial arises in CR geometry 1]. We show that it is the determinant of thep×p circulant matrix whose first row is (1, −x,0,…,0,−y,0,…,0), the −y being in positionq+1. Therefore, the coefficients of this polynomial Φ are integers that count certain classes of permutations. We show that all of the permutations that contribute to a fixed monomialx ^ry^s in Φ have the same sign, and we determine that sign. We prove that a monomialx ^ry^s appears in Φ if and only ifp dividesr+sq. Finally, we show that the size of the largest coefficient of the monomials in Φ grows exponentially withp, by proving that the permanent of the circulant whose first row is (1, 1, 0, …, 0, 1, 0, …, 0) is the sum of the absolute values of the monomials in the polynomial Φ. Supported by NSF Postdoctoral research grants.