Partially directed snake polyominoes

Abstract

Abstract. The goal of this paper is to study the family of snake polyominoes. More precisely, we focus our attention on the class of partially directed snakes. We establish functional equations and length generating functions of two dimensional, three dimensional and then N dimensional partially directed snake polyominoes. We then turn our attention to partially directed snakes inscribed in a b × k rectangle and we establish two-variable generating functions, with respect to height k and length n of the snakes. We include observations on the relationship between snake polyominoes and self-avoiding walks. We conclude with a discussion on inscribed snakes polyominoes of maximal length which lead us to the formulation of a conjecture encountered in the course of our investigations.

The Number of Snakes in a Box

“…In Figure 2.1 we show three, of the 31 quadrilateral snakes of length 7, together with their associated polyomino. We can see that in the third example, the polyomino is not a snake according to our defintion, but it is according to the one used in [3] and [4]. Hence, p(n) actually counts the number of quadrilateral snakes of length n. Therefore, determining a formula for p(n), as well as for p(n), is still an open problem.…”

“…The problem of counting snakes has been considered by several authors. Recently, Goupil et al, [3] studied the problem not only in the plane, they also considered higher dimensions. In that work as well as in [4], the authors accept the third structure in Figure 1.2 as a valid substructure of a snake.…”

Enumeration of parallelogram polycubes

“…Unlike polyominoes, only a few classes of polycubes have been enumerated. Let us cite for instance, the plane partitions [7], the directed plateau polycubes [5], and the partially directed snake polycubes [11]. Some tools were developed for the enumeration of polycubes, in particular the generic method [6], an extension of Bousquet-Melou method [3] and the Dirichlet convolution for the enumeration of polycubes [4].…”

Fully Leafed Induced Subtrees

“…When one adds the constraint of being induced, these walks become thick walks. A particular family of thick walks on the square lattice was successfully investigated in [GPdWd18].…”