Lobb's Generalization of Catalan's Parenthesization Problem

Abstract

where n > 0 [2, 9]. Such numbers are now called Catalan numbers. However, Swiss mathematician Leonhard Euler (1707-1783) had previously encountered them around 1751 while investigating triangulations of convex polygons. In fact, Chinese mathematician Antu Ming (16927-1763?) discovered them even earlier, about 1730, through his geometric models. To our delight and at same time surprise, Catalan numbers occur in numerous seemingly unrelated places and situations, including combinatorics, abstract algebra, linear algebra, algebraic geometry, and sports [9]. They have same delightful propensity for popping up unexpectedly, particularly in combinatorial problems, Mar tin Gardner wrote in 1976 in his popular column Mathematical games in Scientific American [5]. Indeed, he adds, the Catalan sequence is probably most fre quently encountered sequence that is still obscure enough to cause mathematicians lacking access to N. J. A. Sloane's A Handbook of Integer Sequences to expend inor dinate amounts of energy re-discovering formulas that were worked out long ago. For example, Stanley lists over 70 occurrences of Catalan numbers in his book [11] and another 70 on his website Catalan Addendum.