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

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, Ŕt (y; κ) = ∑ m 0 ŕ W t (y;m)κm where ŕt (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 út (y; κ1) ≡ Ŕt (y; κ1 +1) = ∑ m 0 ú W t (y;m)κm 1 where út (y;m) is the number of marked configurations having at least m returns, just m of which are marked. Both ŕt (y;m) and út (y;m) are expressed in terms of the numbers of paths ignoring returns but introducing a suitably modified endpoint condition. This enables út (y;m) to be written in product form for arbitrary y, but for ŕt (y;m) this can only be done in the case y = 0. PACS numbers: 05.50.+q, 05.70.fh, 61.41.+e