ARC Centre of Excellence for Mathematics and Statistics of Complex Systems,
Department of Mathematics and Statistics,
The University of Melbourne,
Parkville, Victoria 3010,
Australia

*e-mail:* `tonyg@ms.unimelb.edu.au`

**Abstract.** Column-convex polyominoes were introduced in 1950's by Temperley,
a mathematical physicist working on ``lattice gases''. By now, column-convex polyominoes
are a popular and well-understood model. There exist several generalizations of column-convex polyominoes.
However, the enumeration by area has been done for only one of the said generalizations,
namely for *multi-directed animals*. In this paper, we introduce a new sequence
of supersets of column-convex polyominoes. Our model
(we call it *level* *m* *column-subconvex polyominoes*) is defined in a simple way:
every column has at most two connected components and, if there are two connected components,
the gap between them consists of at most *m* cells. We focus on the case when cells are
hexagons and we compute the area generating functions for the levels one and two.
Both of those generating functions are *q*-series, whereas the area generating function of
column-convex polyominoes is a rational function. The growth constants of level one and level
two level two column-subconvex polyominoes are 4.319139 and 4.509480, respectively.
For comparison, the growth constants of column-convex polyominoes, multi-directed animals
and all polyominoes are 3.863131, 4.587894 and 5.183148, respectively.

**2000 Mathematics Subject Classification.**
05B50, 05A15.

**Key words and phrases.** Polyomino, hexagonal cell, nearly convex column, area generating function, growth constant.

DOI: 10.3336/gm.45.2.03

**References:**

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MathSciNet CrossRef - M. Bousquet-Mélou, Rapport Scientifique pour obtenir l'habilitation \`a diriger des recherches, Report No. 1154-96, LaBRI, Université Bordeaux I, 1996.
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MathSciNet CrossRef - S. Feretić,
*Polyominoes with nearly convex columns: A semi-directed model,*Ars Math. Contemp., submitted, arXiv:0910.4573v2. - S. Feretić and A. J. Guttmann,
*Two generalizations of column-convex polygons,*J. Phys. A**42**(2009), no. 48, 485003, 17 pp.

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MathSciNet CrossRef - Polygons, Polyominoes and Polycubes (ed. A. J. Guttmann), Lecture Notes in Phys.
**775**, Springer, Berlin, 2009. -
`http://www.gradri.hr/adminmax/files/staff/A_2.mw`(This file is a*Maple 9.5*worksheet. The file can also be obtained from Svjetlan Feretić via e-mail.)