The Enumeration of Prudent Polygons by Area and its Unusual Asymptotics

Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and \emph{$k$-sided prudent walks} (with $k=1,2,3,4$) are, in essence, only allowed to grow along $k$ directions. Prudent polygons are prudent walks that return to a point adjacent to their starting point. Prudent walks and polygons have been previously enumerated by length and perimeter (Bousquet-M\'elou, Schwerdtfeger; 2010). We consider the enumeration of \emph{prudent polygons} by \emph{area}. For the 3-sided variety, we find that the generating function is expressed in terms of a $q$-hypergeometric function, with an accumulation of poles towards the dominant singularity. This expression reveals an unusual asymptotic structure of the number of polygons of area $n$, where the critical exponent is the transcendental number $\log_23$ and and the amplitude involves tiny oscillations. Based on numerical data, we also expect similar phenomena to occur for 4-sided polygons. The asymptotic methodology involves an original combination of Mellin transform techniques and singularity analysis, which is of potential interest in a number of other asymptotic enumeration problems.Comment: 27 pages, 6 figure

The number of directed k-convex polyominoes

We present a new method to obtain the generating functions for directed convex polyominoes according to several different statistics including: width, height, size of last column/row and number of corners. This method can be used to study different families of directed convex polyominoes: symmetric polyominoes, parallelogram polyominoes. In this paper, we apply our method to determine the generating function for directed k-convex polyominoes. We show it is a rational function and we study its asymptotic behavior

Enumeration of Fuss-skew paths

In this paper, we introduce the concept of a Fuss-skew path and then we study the distribution of the semi-perimeter, area, peaks, and corners statistics. We use generating functions to obtain our main results

Minimal Inscribed Polyforms

A polyomino of size n is constructed by joining n unit squares together by their edge to form a shape in the plane. This thesis will first examine the formal definition of a polyomino and the common equivalence classes polyominos are enumerated under. We then turn to polyomino families, and provide exact enumeration results for certain families, including the minimal inscribed polyominos. Next we will generalize polyominos to polyforms, and provide novel formulae for polyform analogues of minimal inscribed polyominos. Finally, we discuss some further questions concerning minimal inscribed polyforms