Partially directed snake polyominoes

Abstract

We introduce and investigate the family of partially directed snake polyominoes. We first present a classification of snake polyominoes and then we turn our attention on the class of partially directed snakes. We establish recurrences, functional equations and generating functions with respect to length and height for two dimensional, three dimensional and d dimensional partially directed snake polyominoes. We then inscribe partially directed snake polyominoes in a b×h rectangle. In order to include the third variable b in our study, we trade the parameter b for two variables w,v to obtain recurrences and functional equations with four variables. We investigate subfamilies of partially directed snake polyominoes. One of these families extends to Zd. In particular, we show a bijective relation between the subfamily of bubble polyominoes and bargraph polyominoes.