A q-enumeration of convex polyominoes by the festoon approach

Authors:
Svjetlan Feretić
Affiliations:
Setaliste Joakima Rakovca 17, 51000 Rijeka, Croatia
Venue:
Theoretical Computer Science - Combinatorics of the discrete plane and tilings
Year:
2004

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Abstract

In 1938, Pólya stated an identity involving the perimeter and area generating function for parallelogram polyominoes. To obtain that identity, Pólya presumably considered festoons. A festoon (so named by Flajolet) is a closed path w which can be written as w = uv, where each step of u is either (1, 0) or (0, 1), and each step of v is either (-1, 0) or (0, -1).In this paper, we introduce four new festoon-like objects. As a result, we obtain explicit expressions (and not just identities) for the generating functions of parallelogram polyominoes, directed convex polyominoes, and convex polyominoes.