ALOIS PANHOLZER AND HELMUT PRODINGER

Transcription

1 ASYMPTOTIC RESULTS FOR THE NUMBER OF PATHS IN A GRID ALOIS PANHOLZER AND HELMUT PRODINGER Abstact. In two ecent papes, Albecht and White, and Hischhon, espectively, consideed the poble of counting the total nube P,n of cetain esticted lattice paths in an n gid of cells, which appeaed in the context of counting tain paths though a ail netwok. Hee we give a pecise study of the asyptotic behaviou of these nubes fo the squae gid, extending the esults of Hischhon, and futheoe povide an asyptotic equivalent of these nubes fo a ectangula gid with a constant popotion α /n between the side lengths.. Intoduction In the ecent pape [], Albecht and White studied the poble of counting the nube P,n of lattice paths in an n ectangula gid stating at, p and ending at, q, fo soe p, q with p q n, whee the peissible oves fo i, j ae to i, j +, i +, j o i +, j + ; thoughout this pape we speak then about peissible lattice paths. These nubes aise in connection with a scheduling poble fo tain paths in a ail netwok see [], whee, as the authos ention, the ode of agnitude of the nubes P,n fo lage values of and n would be of inteest. An exaple visualizing all such paths fo and n 3 is given in Figue. Figue. A visualization of all 4 peissible lattice paths fo a gid of ows and n 3 coluns; thus P,3 4. By solving a bivaiate ecuence via geneating functions, in [] an explicit foula fo P,n has been obtained. Recently, Hischhon [6] obtained the following siple expession fo P,n : P,n n + k. k k + k 0 00 Matheatics Subject Classification. Piay 05A6; Seconday 05A5. Key wods and phases. esticted lattice paths, asyptotic enueation, diagonalization ethod, saddle point ethod.

2 A. PANHOLZER AND H. PRODINGER Using this foula and ealie esults fo [5], Hischhon [6] could descibe the asyptotic behaviou of P,n fo the paticula case n, i.e., the diagonal eleents, as follows: +, P, + fo. 6π Based on nueical coputations he also ade the following conjectue about the second ode te in the asyptotic expansion of P, : + P, + c 6π + o, fo, 3 with a constant c The ai of this pape is to give a oe detailed study of the asyptotic behaviou of the nubes P,n fo lage values of and n, as it is of inteest hee. Fist we deonstate how to pove the conjectue 3 and to identify the appeaing constant c fo the diagonal eleents P,. The ethod applied would, at least in pinciple, even allow to obtain asyptotic expansions fo P, of abitay high ode; hee we estict ouselves to state the fist thee tes in the asyptotic expansion, but one could easily go futhe. Secondly, we povide esults fo the asyptotic behaviou of P,n fo, n, if αn with a positive constant α R +, which coves the ost ipotant gowth ange of and n. One could obtain also efined esults and extensions to othe gowth anges of and n, but we estict ouselves to this case, since it sees to be of ost inteest and we want to avoid unning into futhe technicalities. We eak that thee ae elations to the poble of counting the nube, of paths a king can coss an chessboad fo the lowe left hand cone to the uppe ight hand cone by using only oves to a neighbouing squae eithe to the ight o upwads o diagonally upwads to the ight, which has been studied by Hischhon in [5]. As a consequence of ou coputations we also get a efineent on the coesponding asyptotic esults stated in [5]. Ou esults ae obtained by applying coplex analytic techniques, naely the socalled diagonalization ethod [4] and the saddle point ethod see, e.g., [3], espectively, and use as a stating point the following explicit foula fo the bivaiate geneating function P x, z :,n P,nx z n, which has been coputed in [, 6]: xz P x, z z x z xz. 4. Asyptotic esults fo the diagonal eleents We apply a ethod intoduced by Hautus and Klane [4] in cobination with singulaity analysis of geneating functions to obtain pecise esults concening the asyptotic behaviou of the diagonal eleents P,, i.e., the nube of peissible lattice paths in an squae gid, fo. Let us assue that the bivaiate geneating function F x, z :,n 0 F,nx z n of a sequence F,n conveges fo all x and z such that z < A and x < B, fo abitay A, B > 0. Then it has been shown in [4] that, fo all coplex t with t < AB,

3 ASYMPTOTIC RESULTS FOR THE NUMBER OF PATHS IN A GRID 3 the geneating function ˆF t : n 0 F n,nt n of the diagonal can be coputed via the following contou integal: ˆF t F t, z z dz, 5 πi C z whee the contou C is a siple closed positively oiented cuve aound the oigin staying in the annulus {z C : t < z < A}. B Consideing the bivaiate geneating function P x, z of the nubes P,n as given in equation 4 it is iediate to see that the seies cetainly conveges fo all x, z with x, z <. Thus, fo all coplex t with t <, the geneating function 3 ˆP t : P,t of the diagonal eleents can be obtained by the contou integal ˆP t πi C P t, z z dz, z whee we can always choose as contou C a positively oiented cicle aound the oigin with adius 3. Plugging in 4 we get afte siple anipulations 0 ˆP t P t, z z dz t dz. 6 πi z πi z z tz + t C The solutions of the equation z tz + t 0 ae given by z t t 6t + t and z t t + 6t + t. It holds that z t 0 and z t, fo t 0. Thus, fo all t in a coplex neighbouhood of t 0, it holds that in the contou integal 6 the only singulaity enclosed by the cicle C of adius 3 is a siple pole at z z 0 t. Thus we can evaluate the contou integal 6 by an application of the esidue theoe and obtain the following epesentation of the geneating function ˆP t of the diagonal eleents P, valid a pioi in a coplex neighbouhood of t 0, but which is uniquely given in a uch lage coplex doain due to analytic continuation: t ˆP t Res zz t z z z tz z t t z t z t z t 4t + t + 6t + t 6t + t t t + + t 6t + t 6t + t t + t 6t + t 6t + t 8t t 8t 6t + t + t 8t. 7 As we have eaked ealie thee ae elations between the poble consideed and counting the nube, of paths a king can coss an chessboad fo a cone of the boad to the opposite one, whee only the thee kind of fowad oves ae C

4 4 A. PANHOLZER AND H. PRODINGER peissible; we ae now going to ake this elation pecise. It has been stated in [5, ] that the geneating function Rt : 0 +,+t is given by Rt 6t + t, 8 which iplies 8t ˆP t t Rt + t. Extacting coefficients shows then the elation P, 8,, fo, 9 whee, as usual, denotes the fowad diffeence opeato, i.e., f : f + f, fo an abitay function f. This iplies also that, is divisible by 8. Since it ight be of independent inteest, we fist deduce fo Rt an asyptotic expansion fo the nubes +,+ and then use 9 to show a coesponding one fo P,. We get 6t + t t t t t, with : 3 + and : 3. Thus the unique doinant singulaity, i.e., the singulaity of sallest odulus, of Rt is at t. Accoding to singulaity analysis see [3] the asyptotic behaviou of the coefficients of Rt t t is deteined by the local behaviou of Rt in a coplex neighbouhood of the doinant singulaity t. Thus we expand Rt aound t i.e., in tes of t, whee we estict ouselves to deteine the fist thee tes in the asyptotic expansion. We get then Rt t t t t t + t + t t + 3 t 8 + O t 3 t t t O 0 t 5. Via extacting coefficients fo the binoial seies and applying singulaity analysis, espectively, we get fo 0 iediately +,+ [t ]Rt O , with 3 + and α : α α α α + the coon definition of the! binoial coefficient fo α eal and a non-negative intege. To get a final esult we equie an asyptotic expansion of binoial expessions, with s R fixed, fo, which can be obtained easily by using Stiling s +s

5 ASYMPTOTIC RESULTS FOR THE NUMBER OF PATHS IN A GRID 5 foula fo the factoials see, e.g., [] and note that oden copute algeba systes know these expansions; one gets + s s Γs + ss + ss + s 3s O 3. 4 The following theoe easily follows fo equations and and evaluations of the Γ-function. Theoe. The nube +,+ of paths a king can coss an + + chessboad fo a cone of the boad to the opposite one, whee only the thee kind of fowad oves ae peissible adits, fo, the following asyptotic expansion: +, π O Finally, using 9, Theoe leads afte easy coputations also to an asyptotic esult concening the nubes P,. Coollay. The nube P, of peissible lattice paths in an squae gid adits, fo, the following asyptotic expansion: P, + + c 4 4 π + c + O 3, with c and c Asyptotic esults fo ectangula gids with a constant popotion between the side lengths We deteine the asyptotic behaviou of the nube of peissible paths P,n in an n ectangula gid with a constant popotion α /n > 0 between the side lengths and n, fo n, via the saddle point ethod. Fo the bivaiate geneating function P x, z as given in 4 we fist obtain: P,n [x z n ]P x, z [x z n ] z z x + z [x z n ] z [z n + z ] 3 x +z z. + z Due to Cauchy s integal foula we can wite this expession as a contou integal: P,n + z dz : I, 3 πi z n z + C whee C is a positively oiented siple cuve aound the oigin within a suitable coplex doain, e.g., within the punctued unit disc {z C : 0 < z < }. To evaluate the integal expession I asyptotically, we choose the contou C in such a way that

6 6 A. PANHOLZER AND H. PRODINGER it passes though the saddle point z located on the positive eal axis. If we denote the integand of the contou integal by gz : +z, then the saddle point z n z + satisfies g 0. The esulting equation g n + z + z + + z z z n+ z + z n z + z n z +3 has the solutions z, +± + +4nn+3. Thus the saddle point of inteest is n+3 given by nn n + 3 Since we have assued that αn, with a constant α > 0, we can plug this into the pevious expession 4. Afte easy coputations one gets that has the following asyptotic behaviou, fo n : + α α + On. In the pesent poble it suffices that the contou C does not eally pass though the saddle point, but just passes by closely. Since it siplifies the coputations, we choose thus as contou C a positively oiented cicle aound the oigin with adius : + α α, i.e., C {z C : z e iϕ, 0 ϕ π}, with + α α. The idea of the saddle point ethod is that the ain contibution of the contou integal coes fo the cuve in a sall neighbouhood of the saddle point. Theefoe we wite the integal expession as I I + I with I : πi C gzdz and I : πi C gzdz, whee we split the contou C into the following two pats: C : {z C : z e iϕ, ϕ 0 ϕ ϕ 0 }, with ϕ 0 ϕ 0 n n +ɛ and ɛ > 0, C : {z C : z e iϕ, ϕ 0 < ϕ π ϕ 0 }. Fist we evaluate I, which tuns out to give the ain contibution of I, wheeas I is asyptotically negligible. To do this we equie an asyptotic expansion of gz, fo z C, i.e., fo z e iϕ with ϕ sall. Fo +z +e iϕ ++iϕ ϕ +Oϕ3 we easily get + z iϕ + e log + ϕ iϕ + + +Oϕ3 + e + ϕ + + Oϕ3 + Oϕ + Oϕ + Oϕ iϕ ϕ + + ϕ + +Oϕ3 + e + iϕ e + ϕ + Oϕ + Oϕ 3. Analogously one obtains fo z e iϕ iϕ+ ϕ +Oϕ3 the expansion z + e iϕ e ϕ + Oϕ + Oϕ 3.

7 ASYMPTOTIC RESULTS FOR THE NUMBER OF PATHS IN A GRID 7 Togethe with z n n e inϕ we obtain the following local expansion of gz, fo z e iϕ with ϕ sall: gz Using α n + eiϕ n n e + ϕ and + α α one can easily check that + + n 0. This shows that fo all z e iϕ C the following expansion holds: gz + n + e + + ϕ + Oϕ + Oϕ 3. ϕ + On +3ɛ. 5 Equation 5 leads to the following asyptotic evaluation of the integal expession I, fo n : I gzdz + ϕ0 e ϕ πi C π n dϕ ϕ 0 and by using the substitution ϕ t n futhe to I + π n + n π + n + n n ɛ e n ɛ e The integal appeaing in 6 can be evaluated easily by using π e qt dt, fo q > 0, q leading to I π + n + Eventually, siple anipulations yield the expession I + π + n + + n t dt n t dt. 6. +, fo n. 7 To show that the eaining integal expession I is asyptotically negligible one has to conside the integand gz fo z C, i.e., fo z e iϕ with ϕ 0 < ϕ π ϕ 0. It is easily seen that fo z C we get the following bound on gz: gz + eiϕ 0 n e iϕ 0 + O n + e iϕ 0 e iϕ 0.

8 8 A. PANHOLZER AND H. PRODINGER Via standad anipulations, which ae oitted hee, one can show + e iϕ 0 + e iϕ 0 O + e ϕ 0. Since ϕ 0 n +ɛ and αn, we get thus fo all z C the bound + gz O n e βnɛ, with the constant β α + > 0. Thus it also holds I gzdz + πi O C n e βnɛ, 8 which is exponentially sall copaed to I. Theefoe we get P,n I I +I I, which poves the following theoe. Theoe. The nube P,n of peissible lattice paths in an n ectangula gid adits, fo αn, with α > 0 fixed and n, the following asyptotic equivalent: + P,n π + n + + α n π α + n, α with α + α. Refeences [] M. Abaovitz and I. A. Stegun, Handbook of atheatical functions with foulas, gaphs, and atheatical tables, National Bueau of Standads Applied Matheatics Seies, 55, Washington, 964. [] A. R. Albecht and K. White, Counting paths in a gid, The Austalian Matheatical Society. Gazette, 35, 43 48, 008. [3] P. Flajolet and R. Sedgewick, Analytic cobinatoics, Cabidge Univesity Pess, Cabidge, 009. [4] M. L. J. Hautus and D. A. Klane, The diagonal of a double powe seies, Duke Matheatical Jounal, 38, 9 35, 97. [5] M. D. Hischhon, How any ways can a king coss the boad?, The Austalian Matheatical Society. Gazette, 7, 04 06, 000. [6] M. D. Hischhon, Coent on Counting paths in a gid, The Austalian Matheatical Society. Gazette, 36, 50 5, 009. Alois Panholze, Institute of Discete Matheatics and Geoety, Vienna Univesity of Technology, Wiedne Haupst. 8-0/04, 040 Wien, Austia. E-ail addess: Alois.Panholze@tuwien.ac.at Helut Podinge, Matheatics Depatent, Stellenbosch Univesity, 760 Stellenbosch, South Afica. E-ail addess: hpoding@sun.ac.za