AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 81(3) (2021), Pages 447–457 Enumerations on polyominoes determined by Fuss-Catalan words Toufik Mansour Department of Mathematics, University of Haifa 3498838 Haifa, Israel tmansour@univ.haifa.ac.il Jos ´ e L. Ram ´ ırez Departamento de Matem´ aticas Universidad Nacional de Colombia, Bogot´ a Colombia jlramirezr@unal.edu.co Abstract In this paper we introduce the concept of s-Fuss-Catalan words. This new family of words generalizes the Catalan words (taking s = 1), which are a particular case of growth-restricted words. Here we enumerate the polyominoes or bargraphs associated with the s-Fuss-Catalan words ac- cording to the semiperimeter and area statistics. Additionally, we obtain combinatorial formulas to count the s-Fuss-Catalan bargraphs according of these statistics. 1 Introduction Given a positive integer s, an s-Fuss-Catalan path of length (s + 1)n is a lattice path in the first quadrant of the xy -plane from (0, 0) to the point ((s + 1)n, 0) using up-steps U s = (1,s) and down-steps D = (1, -1). For s = 1 we recover the concept of the classical Dyck path of length 2n enumerated by the famous Catalan numbers C n = 1 n+1 ( 2n n ) . The number of s-Fuss-Catalan paths of length (s + 1)n is given by the Fuss-Catalan numbers C n,s = 1 sn+1 ( (s+1)n n ) . There are several combinatorial interpretations for both the Catalan numbers and for Fuss-Catalan numbers (see, for example, [14] and [9]). For an s-Fuss-Catalan path of length (s + 1)n, we associate the word formed by the subtraction s - 1 from the y -coordinate of each final point of the U s steps. This family of words is called s-Fuss-Catalan words. See Figure 1 for an example. ISSN: 2202-3518 c The author(s). Released under the CC BY 4.0 International License