A note on super Catalan numbers
aa r X i v : . [ m a t h . C O ] M a y A note on super Catalan numbers
Evangelos Georgiadis ∗ Akihiro Munemasa † Hajime Tanaka ‡ May 22, 2012
Abstract
We show that the super Catalan numbers are special values of the Krawtchoukpolynomials by deriving an expression for the super Catalan numbers in terms of asigned set.
The super Catalan numbers, S ( m, n ) := (2 m )!(2 n )! m ! n !( m + n )! , as designated by Gessel in [9, Eq. (28)] form, hierarchically speaking, special cases of superballot numbers (cf. [9, pp. 180, 189]). Historically, Gessel points out that these numbers hadbeen observed as early as 1874 and studied by E. Catalan [6]; Aguiar and Hsiao in [1] providea more detailed account of earlier appearences (cf. [5], [8], [7] and Riordan [14, Chapter 3,Exercise 9, p. 120]). Surprisingly, the innocent looking numbers, S ( m, n ), seem to havebeen immune against a combinatorial interpretation for all values of ( m, n ) over the lastcentury despite limited success stories for particular values (see problem 66(a) in Stanley’sbijective open problems compendium [16]). The following references highlight the successcases. In [9], Gessel notes that for S (1 , n ) / C n ; whereasfor the case when m = 0, we yield middle binomial coefficients, (cid:0) nn (cid:1) . In [10], Gessel andXin provide a combinatorial interpretation in terms of Dyck paths when m = 2 or 3. Analternative combinatorial interpretation for the case m = 2 was provided by Schaeffer in [15]using a method that was introduced in the interpretation to formulas of Tutte for planarmaps. A more topologically flavored yet still combinatorial interpretation for the m = 2 caseis also available by Pippenger and Schleich in [13]; they count cubic trees on n interior vertices(or the number of hexagonal trees with n nodes). In 2005, Callan in [4] provided an elegant ∗ Massachusetts Institute of Technology, Cambridge, M.A. 02139, U.S.A.
Email: [email protected] † Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan.
Email: [email protected] ‡ Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan.
Email: [email protected] S ( m, n ) / P k ≥ n − m − k (cid:0) n − m k (cid:1) S ( m, k ) / m = 2 showing that it enumerates the aligned cubic trees by number ofvertices that are neither a leaf nor adjacent to a leaf.In this note we establish the following expression for super Catalan numbers: S ( m, n ) = ( − m X P ∈P m + n ( − h m ( P ) , (1)where the sum is over the set P m + n of all lattice paths from (0 ,
0) to ( m + n, m + n ) consisting ofunit steps to the right and up, and h m ( P ) denotes the height of P = ( P , P , . . . , P m + n ) ) ∈P m + n after the 2 m th step, i.e., the y -coordinate of P m . Although this is an interpretationof S ( m, n ) in terms of a signed set only, the right-hand side of (1) is a special value of theKrawtchouk polynomial defined as follows: K dj ( x ) = j X h =0 ( − h (cid:18) xh (cid:19)(cid:18) d − xj − h (cid:19) . Then (1) is equivalent to K m + n ) m + n (2 m ) = ( − m S ( m, n ) . (2)To see the equivalence, observe that each P ∈ P m + n has exactly m + n up-steps and thatthe number of P ∈ P m + n with h m ( P ) = h is therefore equal to (cid:0) mh (cid:1)(cid:0) nm + n − h (cid:1) .Krawtchouk polynomials K dj ( x ) appear as the coefficients of the so-called MacWilliamsidentities (cf. [12, Chap. 5, § j graph of the d -cube (cf. [3, Chap. 3, § { ( − m S ( m, n ) | m, n ≥ , m + n = N } coincides with the set of non-zero eigenvalues of the distance- N graph of the 2 N -cube,which is known as the orthogonality graph and has been studied in connection with pseudo-telepathy in quantum information theory (cf. [11]). Finally, (2) follows immediately fromthe identity of von Szily (cf. [9, Eq. (29)]): S ( m, n ) = X k ∈ Z ( − k (cid:18) mm + k (cid:19)(cid:18) nn − k (cid:19) = ( − m m + n X h =0 ( − h (cid:18) mh (cid:19)(cid:18) nm + n − h (cid:19) = ( − m K m + n ) m + n (2 m ) . We note that (2), as well as the identity of von Szily, is just a restatement of (a special caseof) Kummer’s evaluation of well-poised F ( −
1) series.To obtain a proper interpretation as the size of a set of certain paths, we need to find aninjection from the set { P ∈ P m + n | h m ( P ) m (mod 2) } to { P ∈ P m + n | h m ( P ) ≡ m (mod 2) } , and a description of the complement of the image. This is known for the case m = 1 (see [2,Section 5.3]), but it seems to be a difficult problem in general.2 cknowledgments Special thanks are due to Ole Warnaar and Richard Askey for valuable comments. E.G. isindebted to M. Sipser and J. Tsitsiklis of MIT and would like to thank I. Gessel of Bran-deis University for helpful background comments on this problem in early 2010. H.T. wassupported in part by the JSPS Excellent Young Researchers Overseas Visit Program.
References [1] M. Aguiar and S. K. Hsiao, Canonical characters on quasi-symmetric functions and bi-variate Catalan numbers, Electron. J. Combin. 11 (2004/06) R15; arXiv:math/0408053.[2] M. Aigner, A Course in Enumeration, Springer, 2007.[3] E. Bannai and T. Ito, Algebraic Combinatorics, I: Association Schemes, Ben-jamin/Cummings, Menlo Park, 1984.[4] D. Callan, A combinatorial interpretation for a super-Catalan recurrence, J. IntegerSeq. 8 (2005) Article 05.1.8; arXiv:math/0408117.[5] E. Catalan, Sur quelques questions relatives aux fonctions elliptiques, Seconde Note.Pr´esent´ee `a l’Acad´emie pontificale des Nuovi Lincei dans la s´eance du 19 Janvier 1873.[6] E. Catalan, Question 1135, Nouvelles Annales de Math´ematiques: Journal des Candi-dats aux ´Ecoles Polytechnic et Normale, Series 2, 13, 207.[7] E. Catalan, M´elanges Math´ematiques, Tome II, Bruxelles, F. Hayez, 1887. Publishedalso in Extrait des M´emoires de la soci´et´e royale des sciences de Li´ege, 2e s´er., XIII,Paris, Gauthier-Villars, 1885.[8] E. Catalan, M´emoire sur quelques d´ecompositions en carr´es, Atti dell’ Accademia Pon-tificia Romana de Nuouvi Lincei, v. XXXVII, sessione I (1883), 49–114.[9] I. M. Gessel, Super ballot numbers, J. Symbolic Comput. 14 (1992) 179–194.[10] I. M. Gessel and G. Xin, A combinatorial interpretation of the numbers 6(2 n )! /n !( nn