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Transcript
^040# THE PARITY OF THE CATALAN NUMBERS VIA LATTICE PATHS OMER EGECIOGLU University of California San Diego, La Jolla, (Submitted April 1982) CA 92093 The Catalan numbers Cn = (2nn)/(n + l) belong to the class of advanced counting numbers that appear as naturally and almost as frequently as the binomial coefficients$ due to the extensive variety of combinatorial objects counted by them (see [1]9 [2])« The purpose of this note is to give a combinatorial proof of the following property of the Catalan sequence using a lattice path interpretation. Theorem Cn is odd if and only if n = 2r - 1 for some positive integer v. Proof: The proof is based mainly on the following observation: If X is a finite set and a is an involution on X with fixed point set Xas then |z| = | j a | (mod 2); i.e.s \x\ and | j a | have the same parity,, Now let Dn denote the set of lattice paths in the first quadrant from the origin to the point (2ns 0) with the elementary steps x : (a, b) -»• (a + 1, b + 1) (a, b) -*- (a + 1, b - 1). xi It is well known that \Dn\ = Cn (see [2], [3]). Define a : Dn -* Dn by reflecting these paths about the line x = n* The fixed point set D% of a consists of all paths in Dn symmetric with respect to the line x - n. Now define an involution 3 on D% as follows: for w-W^uuWi w \ i\ = 1^21 = n ~~ * and Of course the set Dn-i u e x {* ^ 3 &% with set 1 w1uuw2 if I otherwise. w e ^i 4 Dn-i is empty unless n is odd. Hence9 we can put 2 Cn-i ~ 0 f° r n even. 2 19S3J 65 THE PARITY OF THE CATALAN NUMBERS VIA LATTICE PATHS Note that \K 2I since w -*• W1 is an obvious bijection between the sets D* we have £n = Cn-i and Dn-i. 2 (mod 2). Thus (1) 2 If Cn is odd, then induction on n gives (n - l)/2 = 2r - 1 for some r so that n - 2r+1 - 1 is of the required form. Of course, C2-i ~ ^1 = ^' The converse also follows immediately from (1) by a similar inductive argument. REFERENCES 1. H. W. Gould. "Research Bibliography of Two Special Number Sequences." Mathematicae Monogaliae (Dept. of Math., W.Va. Univ. 26506), 1971. 2. J. H. van Lint. Combinatorial Theory Seminar: Lecture matics No. 382. New York: Springer-Verlag, 1974. 3. W. Feller. An Introduction to Probability tions. New York: John Wiley & Sons, 1950. Notes Theory and Its in MatheApplica- • 0*0* 66 [Feb.