Some remarks on Catalan families (Q1100466)

The author presents a simple direct approach to the main results on Catalan families (the number of r-ary trees with n points and precisely \(t_ i\) i-edges, the number of nonnegative paths from (0,0) to (rn,0) with \(k+1\) peaks). His starting point is the problem of counting the number of all possible products formed from \((r-1)n+1\) different elements using an r-ary nonassociative noncommutative multiplication. The usual Catalan families are obtained by identifying all elements from \(\{\) 0,1,...,(r-1)n\(\}\).