Return polynomials for non-intersecting paths above a surface on the directed square lattice

Abstract

We enumerate sets of n non-intersecting, t-step paths on the directed square lattice which are excluded from the region below the surface y = 0 to which they are initially attached. In particular we obtain a product formula for the number of star configurations in which the paths have arbitrary fixed endpoints. We also consider the 'return' polynomial, ŔWt (y; K) = ∑m≥0 ŕWt (y; m) Km where ŕWt (y; m) is the number of n-path configurations of watermelon type having deviation y for which the path closest to the surface returns to the surface m times. The 'marked return' polynomial is defined by úWt(y; K1) ≡ ŔWt (y; K1 + 1) = ∑m≥0 úWt (y; m) Km1 where úWt (y; m) is the number of marked configur..