INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL J. Phys. A: Math. Gen. 35 (2002) 3213–3230 PII: S0305-4470(02)31988-7 Anisotropic step, mutual contact and area weighted festoons and parallelogram polyominoes on the triangular lattice A C Oppenheim, R Brak and A L Owczarek Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3052, Australia E-mail: aleks@ms.unimelb.edu.au (A L Owczarek) Received 17 December 2001 Published 29 March 2002 Online at stacks.iop.org/JPhysA/35/3213 Abstract We present results for the generating functions of polygons and more general objects that can touch, constructed from two fully directed walks on the infinite triangular lattice, enumerated according to each type of step and weighted proportional to the area and the number of contacts between the directed sides of the objects. In general these directed objects are known as festoons, being constructed from the so-called friendly directed walks, while the subset constructed from vicious walks are staircase polygons, also known as parallelogram polyominoes. Additionally, we give explicit formulae for various first area-moment generating functions, that is when the area is summed over all configurations with a given perimeter. These results generalize and summarize nearly all known results on the square lattice, since such results can be obtained by setting one of the step weights to zero. All our results for the triangular lattice are new and hence provide the opportunity to study subtle changes in scaling between lattices. In most cases we give our results both in terms of ratios of infinite q-series and as continued fractions. PACS numbers: 05.50.+q, 02.10Ab, 61.41.+e 1. Introduction Directed versions of polyominoes or polygons provide exactly solvable versions of the fundamental lattice models of lattice animals and self-avoiding polygons. They are hence a valuable testing ground for hypotheses concerning these more general models which are themselves lattice models of branched polymers and vesicles [1]. Intriguingly, recent work [2] has argued that the extent to which such directed models can mimic the behaviour of their unrestricted cousins is perhaps greater than is apparent on first glance. Staircase polygons, 0305-4470/02/143213+18$30.00 © 2002 IOP Publishing Ltd Printed in the UK 3213