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^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*
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[Feb.