Svjetlan Feretić and Anthony J Guttmann 2009 J. Phys. A: Math. Theor. 42 485003 doi:10.1088/1751-8113/42/48/485003
Svjetlan Feretić1 and Anthony J Guttmann2
Show affiliationsColumn-convex polygons were first counted by area several decades ago, and the result was found to be a simple, rational, generating function. In this work we generalize that result. Let a p-column polyomino be a polyomino whose columns can have 1, 2, ..., p connected components. Then column-convex polygons are equivalent to 1-convex polyominoes. The area generating function of even the simplest generalization, namely 2-column polyominoes, is unlikely to be solvable. We therefore define two classes of polyominoes which interpolate between column-convex polygons and 2-column polyominoes. We derive the area generating functions of those two classes, using extensions of existing algorithms. The growth constants of both classes are greater than the growth constant of column-convex polyominoes. Rather tight lower bounds on the growth constants complement a comprehensive asymptotic analysis.
02.10.De Algebraic structures and number theory
02.60.Jh Numerical differentiation and integration
05A10 Factorials, binomial coefficients, combinatorial functions (See also 11B65, 33Cxx)
26C05 Polynomials: analytic properties, etc. (See also 12Dxx, 12Exx)
Issue 48 (4 December 2009)
Received 24 June 2009, in final form 6 October 2009
Published 12 November 2009
Svjetlan Feretić and Anthony J Guttmann 2009 J. Phys. A: Math. Theor. 42 485003
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