Cool-lex order and k-ary Catalan structures

For any given k, the sequence of k-ary Catalan numbers, Ct,k = 1 kt+1 ( kt t ) , enumerates a number of combinatorial objects, including k-ary Dyck words of length n = kt and k-ary trees with t internal nodes. We show that these objects can be efficiently ordered using the same variation of lexicographic order known as cool-lex order. In particular, we provide loopless algorithms that generate each successive object in O(1) time. The algorithms are also efficient in terms of memory, with the k-ary Dyck word algorithm using O(1) additional index variables, and the k-ary tree algorithm using O(t) additional pointers and index variables. We also show how to efficiently rank and unrank k-ary Dyck words in cool-lex order using O(kt) arithmetic operations, subject to an initial precomputation. Our results are based on the cool-lex successor rule for sets of binary strings that are bubble languages. However, we must complement and reverse 1/k-ary Dyck words to obtain the stated efficiency.