2020/07/01 by Msapato, Dixy
#05A10 (Primary) #05A19 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2007.00718
The modular Catalan numbers Ck,n, introduced by Hein and Huang in 2016 count equivalence classes of parenthesizations of x0 * x1 * … *xn where * is a binary k-associative operation and k is a positive integer. The classical notion of associativity is just 1-associativity, in which case C1,n = 1 and the size of the unique class is given by the Catalan number Cn. In this paper we introduce modular Fuss-Catalan numbers Ck,nm which count equivalence classes of parenthesizations of x0 * x1 * … *xn where * is an m-ary k-associative operation for m ≥ 2. Our main results are a closed formula for Ck,nm and a characterisation of k-associativity.