Enumeration of area-weighted Dyck paths with restricted height

Transcription

1 AUSTRALASIAN JOURNAL OF COMBINATORICS Volue 54 (2012), Pages Enueration of area-weighted Dyck paths with restricted height A.L. Owczarek Departent of Matheatics and Statistics The University of Melbourne Parkville, Victoria 3010 Australia T. Prellberg School of Matheatical Sciences Queen Mary University of London Mile End Road, London E1 4NS U.K. Abstract We derive explicit expressions for -orthogonal polynoials arising in the enueration of area-weighted Dyck paths with restricted height. 1 Introduction and Stateent of Results Dyck paths are directed walks on Z 2 starting at (0, 0) and ending on the line y =0, which have no vertices with negative y-coordinates, and which have steps in the (1, 1) and (1, 1) directions [11]. We ipose the additional geoetrical constraint that the paths have height at ost h, that is, they lie between lines y =0and y = h. Given a Dyck path π, we define the length n(π) tobehalfthenuberof its steps, and the area (π) to be the su of the starting heights of all steps in the (1, 1) direction in the path. Actually, (π) isthevalueofπ in the classical lattice of Dyck paths [4, 9], and is euivalent to the nuber of diaond plauettes under the Dyck path. An alternative definition of the area is the su of the heights of all steps in the Dyck path, which evaluates to n(π) + 2(π) and is euivalent to the nuber of triangular plauettes under the Dyck path. The definition used here has the advantage of enabling a ore elegant and concise atheatical forulation of our results. We denote by u(π) andv(π) the nuber of vertices in the line y = 0 (excluding the vertex (0, 0)) and the nuber of vertices in the line y = h, respectively. Let D h

2 14 A.L. OWCZAREK AND T. PRELLBERG be the set of Dyck paths with height at ost h, and define the generating function D h (a, b;,t) = a u(π) b v(π) (π) t n(π). π D h The generating function for Dyck paths with restricted height [2, 5] and the generating function for area-weighed Dyck paths [3, 5] have previously been studied. Here we extend these works by cobining these properties. The purpose of this note is to derive the following expression for D h (a, b;,t). Theore 1. For h 0, D h (a, b;,t) = ( t) ( 1) (1 b)[ h ] +b[ h+1 ] (1 b)[ h 1 1 ] b[ h 1] ( t) ( 1) (1 b)[ h ] +b[ h+1 ] (1 a)(1 b)[ h 1 1 ] (1 a)b[ h 1]. (1) Here, we have used the standard notation for -binoial coefficients [8, 10]; for n 0and0 k n we define [ ] n 1 n (; ) n =, where (a; ) n = ( a i ). k (; ) k (; ) n k i=0 We extend this definition to integer-valued n, k by letting [ ] n =0whenk<0or k k>n. (This definition is coensurate with the lattice path definition of -binoial coefficients and is necessary to allow for Theore 1 to be valid even for h =0.) For a = b = 1, this identity siplifies considerably. Corollary 2. For h 0, [ ( t) 2 h ] D h (1, 1;,t) = ( t) ( 1)[ ] h+1 Taking the liit h, we recover the well-known result [6] that the areaweighted generating function D(,t) for Dyck paths without height restriction is given by ( t) 2 (; ) D(,t) = ( t) ( 1) (; ) More precisely, D(,t) as defined here is related to F (z,) in En. (75) of Chapter Vin[6]viaF (z,) =D(,z)...

3 2 Proofs ENUMERATION OF AREA-WEIGHTED DYCK PATHS 15 We use as the starting point of our derivation a well-established connection between lattice path enueration and continued fractions [5]. Proposition 3. D 0 (a, b;,t) =b, D 1 (a, b;,t) =1/( abt), andforh 2, D h (a, b;,t) = 1 at t 2 t 3 t. (2)... h 2 t b h 1 t While this can easily be proved by specialising the general theory in [5] to the case at hand, we shall give a direct cobinatorial proof. Proof. The only Dyck path of height zero is the zero step Dyck path. If h =0then it has weight b, whence D 0 (a, b;,t) =b. Let now h 1. Except for the zero-step Dyck path with weight 1, every Dyck path of height at ost h can be decoposed uniuely into a Dyck path of height at ost (h 1) bracketed by a pair of steps into the (1, 1) and (1, 1) directions, followed by another Dyck path of height h. The associated generating functions are atd h 1 (1,b;,t) andd h (a, b;,t), respectively. This decoposition leads to the functional-recurrence D h (a, b;,t) =1+atD h 1 (1,b;,t)D h (a, b;,t). Iterating D h (a, b;,t) =1/( atd h 1 (1,b;,t)) gives (2). It is clear that the generating function can also be written as a rational function, and fro Section 3 in [5] we obtain the following three-ter recurrence. Proposition 4. For h 1, D h (a, b;,t) = Q h(0,b;,t) Q h (a, b;,t), where abt, h =1, Q h (a, b;,t) = at bt, h =2, Q h 1 (a, 1;,t) b h 1 tq h 2 (a, 1;,t) h 3. (3)

4 16 A.L. OWCZAREK AND T. PRELLBERG Proof. The initial conditions follow fro D 1 (a, b;,t) =1/( abt) andd 2 (a, b;,t) =(1 bt)/(1 at bt), and the factor b h 1 t in the three-ter recurrence is just the final ter in the continued fraction (2). More precisely, coparing En. (2) with the h-th convergent of the J-fraction on page 152 of [5], we have t = z 2, a 0 = a, a k =1 for k 1, b k = k 1 for 0 k<h, b h = b h 1,andc k =0fork 0. The linear recurrences given on page 152 of [5] then reduce to the recurrence in En. (3). We proceed by considering the generating function of the denoinators Q h (a, b;,t), W (z; a, b;,t) = Q h (a, b;,t)z h. h=1 The next proposition expresses W (z; a, b;,t) in ters of the basic hypergeoetric series φ(z,,t) = 1 φ 2 (;0,z;,t) [7], i.e., Proposition 5. φ(z,,t) = n=0 n(n 1) t n (z; ) n. W (z; a, b;,t) = 1 at tz (z+b bz)(1 φ(z,, tz2 ))+(z+b bz)φ(z,, tz 2 ) b. (4) Proof. The recurrence (3) iplies that W (z; a, b;,t) satisfies a functional euation, W (z; a, b;,t) =(1 abt)z btz 2 + zw(z; a, 1;,t) z 2 btw (z; a, 1;,t). Letting b =1andisolatingW (z; a, 1,,t) gives W (z; a, 1;,t) = z z2 ( at zt) tw(z; a, 1;,t). z z This functional euation has the structure W (z) = A(z) + B(z)W (z) whichcan readily be solved by iteration to give W (z) = n=0 A(n z) n 1 k=0 B(k z). In this way we find ( 1) n z 2n+1 n2 +n t n ( at z n+1 t) W (z; a, 1;,t) = (z; ) n+1 n=0 = at tz at φ(z,, tz 2 )+φ(z,, tz 2 ). tz Inserting this expression into the functional euation gives En. (4). Proposition 6. For h 1, ( [ ] h Q h (a, b;,t) = ( t) ( 1) ( b) [ ] [ ] [ ] ) h +1 h h +b ( a)( b) ( a)b. (5) 1 1

5 ENUMERATION OF AREA-WEIGHTED DYCK PATHS 17 Proof. We obtain Q h (a, b;,t) by extracting the coefficient of z h in W (z; a, b;,t). We expand the -product in the function φ with the help of the -binoial theore (see page 490 of [1]) to obtain φ(z,,tz 2 )=1+ z n=1 [ ] n 1 n(n 1) t n. n 1 Inserting this expansion into (4) and collecting ters with eual powers in z gives En. (5). Theore 1 now follows fro Propositions 4 and 6 and by checking that the expression gives the correct result also for h =0. Acknowledgeents Financial support fro the Australian Research Council via its support for the Centre of Excellence for Matheatics and Statistics of Coplex Systes is gratefully acknowledged by the authors. A.L. Owczarek thanks the School of Matheatical Sciences, Queen Mary, University of London for hospitality. References [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, vol.71ofencyclopedia of Matheatics and its Applications, Cabridge University Press, Cabridge, [2] R. Brak, A. L. Owczarek, A. Rechnitzer and S. Whittington, A directed walk odel of a long chain polyer in a slit with attractive walls, J. Phys. A 38 (2005), [3] P. Duchon, On the Enueration and Generation of Generalized Dyck Words, Discrete Math. 225 (2000), [4] L. Ferrari and R. Pinzani, Lattices of lattice paths, J. Statist. Plann. Inference 135 (2005), [5] P. Flajolet, Cobinatorial aspects of continued fractions, Discrete Math. 41 (1980), [6] P. Flajolet and R. Sedgewick, Analytic Cobinatorics, Cabridge University Press, Cabridge, [7] G. Gasper and M. Rahan, Basic Hypergeoetric Series,vol.96ofEncyclopedia of Matheatics and its Applications, Cabridge University Press, Cabridge, 2004.

6 18 A.L. OWCZAREK AND T. PRELLBERG [8] I. P. Goulden and D. M. Jackson, Cobinatorial Enueration, Wiley, New York, [9] A. Sapounakis, I. Tasoulas and P. Tsikouras, On the Doinance Partial Ordering of Dyck Paths, J. Integer Se. 9 no. 2 (2006), Article [10] R. P. Stanley, Enuerative Cobinatorics, Volue 1, vol.49ofcabridge Studies in Advanced Matheatics, Cabridge University Press, Cabridge, [11] R. P. Stanley, Enuerative Cobinatorics, Volue 2, vol.62ofcabridge Studies in Advanced Matheatics, Cabridge University Press, Cabridge, (Received 17 July 2010; revised 5 Nov 2011, 3 July 2012)