Catalan Traffic at the Beach

Transcription

1 Catala Traffc at the Beach Herch Nederhause Departmet of Mathematcal Sceces Florda Atlatc Uversty, Boca Rato, USA Submtted: March 7, 2002; Accepted: August 2, 2002 MR Subject Classfcatos: 05A5, 05A9 Abstract The ubqutous Catala umbers C ( 2 / ( + occur as t (, the followg table showg the umber of ways to reach the pot (, m o a rectagular grd uder certa traffc restrctos, dcated by arrows m gate gate Ξ 2 gate Ξ start here ψ gate Ξ t (, m We prove ths wth the help of hypergeometrc dettes, ad also by solvg a equvalet lattce path problem O the way we pck up several dettes ad dscuss other kow sequeces of umbers occurrg the Catala traffc scheme, lke the Motzk umbers row m, ad the Tr-Catala umbers,, 3, 2, 55, at the gates Catala Traffc There are 66 problems Staley s book Eumeratve Combatorcs, Vol II [0, Chpt ( 6], about dfferet combatoral structures couted by the Catala umbers C 2 + Oe example (Ballot problem, or Dyck paths s the total cout of lattce paths wth step vectors East ( ad South (, startg at the org, edg o the dagoal y + x 0, ever gog below that le If we thk of the lattce as the streets o a cty map, the dagoal as the beach, we are talkg about the umber of shortest trps betwee two spots at the beach the electroc joural of combatorcs 9 (2002, #R33

2 start here ψ Table : The umber of routes from the startg pot to ay other pot o the beach are the Catala umbers We wll prove that the followg obstacle course, a carefully desged system of detours ad rght turs, aga allows us to vst the dagoal the same (Catala- umber of ways as before, wth the added beeft that we caot get off the dagoal aymore m gate gate Ξ 2 gate Ξ start here ψ gate Ξ t (, m The rules are as follows Table 2: Detours preservg Catala traffc At lattce pots strctly above the le x 2y go N or E ( ad 2 At lattce pots strctly below the le x 2y go S ad E, but eforce rght turs whe comg from the West Note that such a rght tur from (, j to( +,j va ( +,j s equvalet to a SE dagoal step We could smply requre that the step vectors are ad 3 Block out all traffc at the tersectos (2m +,m, except for rght turs for paths from the West 4 O the le, at the gates (2m, m, allow N,S, ad rght tur steps (because of the road block at (2m +,m these steps are equvalet to,, ad The pots o ths le are the oly gates from the upper traffc to the lower traffc Catala traffc geerates other kow umber sequeces besdes the Catala umbers As R Sulake poted out to me, the sequece t (,,, must be the sequece of the electroc joural of combatorcs 9 (2002, #R33 2

3 m t (, m Table 3: Dagoal steps ca replace the rght turs (gates bold Motzk umbers f t (, C I addto, the umbers at the gates are also well kow, ( 3 / (2 + ; they may be vewed as the Tr-Catala umbers C ( the famly c / ((c + of Catalas For refereces see the sequece A00764 the O-Le Ecyclopeda of Iteger Sequeces They were the orgal motvato for devsg the Catala traffc whe I was troduced to them the form of a tes ball sequece by Merl, Sprugol, R, ad Verr, M C [5] Some of ths addtoal materal s compled Secto 5 The three types of umbers that occur the Catala traffc scheme are summarzed the followg table Number a Formula Ge Fucto 0 a t Locato ( Catala C 2 / ( t (, 4t / ( t s ( Tr-Catala C ( 3 Motzk M + k0 ( k k ( + k arcs ( 27t/2 t (2, 3 t ( +, 2t 2 ( t ( 3t(+t Table 4: Iterestg umbers occurg the Catala traffc scheme To prove that the umbers t (, alog the dagoal really are the Catala umbers we wll frst reformulate the problem such that the same backwards dfferece recurso holds at every lattce pot Subsequetly we gve three proofs for the observato t (, C The frst par of proofs show that the followg system of equatos 0 2 ( 3 2 x ad ( 3 x 2 s (uquely solved by x ( C for >0, ad x 0 The frst (ad shortest proof, Secto 3, [ employs the help of Zelberger s algorthm for hypergeometrc fuctos to show that 2 F m, ; m/2 ;4] 3 for postve tegers m The secod proof, 2 Secto 32, s also based o recursve evaluatos of hypergeometrc fuctos, offerg the electroc joural of combatorcs 9 (2002, #R33 3

4 more backgroud materal the form of related dettes (Corollary 3, lke C 2 ( ( 3 C, C ( 3 ( C 2 draw from HW Gould s Improved evaluato of the fte hypergeometrc seres F (, /2; j + [2] (Secto 32 The thrd proof, Secto 4, s based o the kow soluto to the eumerato of lattce paths strctly above les of slope 2 Combg the results of these dfferet proofs establshes dettes lke m /2 ( 2(m ( 2(m + for m ( /2 (expasos (7 ad (7, /2 (expaso (8, ad s C ( m + ( C ( ( 3(m m + 2m ( m C 2 ( ( /2 m + ( C ( ( + 3 ( ( 3 arcs t 2 + /2 ( ( 3/2 t 3 ( + C t 3/ ( t + ( 4t Frst, however, we trasform the Catala traffc at the beach to a problem about lattce paths wth steps, strctly above the le y (x /2 (Secto 3, where the Catala umbers occur aga alog the same dagoal, but wth alteratg sgs They wll also occur whe we cout, -paths strctly above the le y x/2 Such paths have bee eumerated by Gessel [] wth hs probablstc approach Ths ad aother approach vald for slopes of the form /g, whereg s a postve teger, are dscussed Secto 5 2 Utaglg the Detours The umbers t (, m Table 3 ca be recursvely calculated from the gve tal codtos However, two smple chages wll make the recursve calculatos easer I the frst step we splt the gates at (2m, m to two cells, (2m, m ad(2m +,m, duplcatg the cotet, ad shftg the whole matrx of umbers below the le 2y x oe step SE Call the ew etres t (, m We have t (, m t (, m f 2m, t (, m t (,m+f 2m+3, ad we fll the crack wth zeroes, t (2m +,m: t (2m +2,m : 0 for all m 0 From the ew pots (2m, m, (2m +,m, ad the electroc joural of combatorcs 9 (2002, #R33 4

5 Stretchg the lattce (the crack brackets Chagg sgs m [0] [0] [0] [0] [0] t (, m 5 9 d(, m Table 5: Ufyg the recurso (2m +2,m the path ca take the steps ad The cojectured -th Catala umber C has moved dow the dagoal to the posto ( +, I the secod step we chage the sg of the etres odd umbered colums, but oly below the crack; call the ew values d (, m We have d (, m ( t (, m f 2m +2,add (, m t (, m f 2m + We clam that d (, m d (, m + d (,m ( for all m Ths trvally holds for 2m + 2, because t was already true for t (, m, ad we flled the crack wth the rght values If 2m +2ad m the d (, m ( t (, m ( (t (, m ++t (,m+ d (, m + d (,m+ cofrmg ( Our cojecture about the Catala umbers Table 2, t (, C,s equvalet to the cojecture d ( +, ( + C (2 for all 0 We prove ths statemet two dfferet approaches The frst proof, Secto 3, shows that d ( +, ( + C are the uque solutos of a system of equatos assocated to our problem We solve the system two ways, wth ad wthout the help of algebrac computer packages The secod proof terprets the umbers d (, m above the crack as the umber of lattce paths from the org to (, m strctly above the le y (x /2 These umbers are kow; ther polyomal expaso shows that (2 holds Because of the smple recurso ad tal values, there must be may other proofs of (2 However, the challege s fdg a combatoral argumet for the appearace of the oalteratg Catala umbers t (, C the orgal problem the electroc joural of combatorcs 9 (2002, #R33 5

6 3 The Hypergeometrc Fucto Approach The recurso ( s a backwards dfferece recurso, d (, m d (, m d (,m, wth tal values d (0, 0 d (, ( /2 0 for all odd >0 (3 d (, 2 0 for all eve > Because we ca exted d (0,m to the costat for all m Z, wecaextedd (, m to a polyomal d (x ofdegree such that d (m d (, m for all m m Table 6: The polyomal exteso d (m ofd(, m Ay polyomal sequece (s (x 0 that satsfes the backwards dfferece equato s (x s (x s (x ca be expaded as s (x s (u + v x u v ( +x u v (4 x u v for ay choce of the scalars u ad v [7, Corollary 23] Apply ths expaso to the case u adv 0toseethat ( + x d (x d ( (5 The tal values (3 mply that for all postve tegers 2 ( d ( ad 0 d ( 2 ( 3 (6 2 Ths s a system of equatos for the ukow coeffcets d ( The followg proposto shows that d ( ( C are the uque solutos to the system f d 0 (0 Ths proves statemet (2 Expaso (5 wll the mply that ( x + d (x + j ( j C j ( x + j, (7 the electroc joural of combatorcs 9 (2002, #R33 6

7 hece we obta the geeratg fucto d (x t 3 t +3t (8 2( t x+3/2 0 Proposto For postve tegers holds ( ( ( ( ad (9 2 2 ( 3 2 ( ( ( (0 2 2 Equvaletly, 2F [ 2 +, /2; ;4] 2 F [ 2, /2; ;4]3 ( Recall that 2 F [a, b; c; z] + a(a+ (a+ b(b+ (b+ z See Subsectos 3! c(c+ (c+ ad 32 for two dfferet proofs of the proposto 3 Proof by Zelberger s algorthm Let f ( 2 F [ 2, /2; ;4] ad g ( 2 F [ 2 +, /2; ; 4] Zelberger s algorthm as mplemeted Maple VII shows that f ( 2( f ( 9( ( 2 f ( 2 ad (3 (3 2 g ( 2( g ( 9( 2 2 g ( 2 (3 2 (3 4 Both recursos are solved by costat fuctos It s easy to check that f ( g ( 3, hece f ( g ( 3 for all postve tegers Ths proves Proposto the form of equato ( 32 Proof by Gould s specal hypergeometrc fucto HW Gould [2] vestgated the hypergeometrc fucto 2 F [, /2; m;4z] for tegers m ad postve tegers Gould foud tegral represetatos ad recursos volvg 2F [ 2, /2; m;4] ad F [ 2 +, /2; m; 4] Importat for ths paper are hs evaluatos of the specal cases [3, (733-(735,(740-(742] 2F [ 2, /2; ;4]3, 2F [ 2 +, /2; +;4]0, (2 2F [ 2 +, /2; ;4] 2F [ 2, /2; +2;4] + 2 +, (3 2F [ 2, /2; +;4], 2F [ 2 +, /2; +2;4] + 2(2 + (4 We also eed the followg techcal lemma the electroc joural of combatorcs 9 (2002, #R33 7

8 Lemma 2 Let ad m be tegers, >0 ad m 0 The 2F [, /2; m;4z] 2 F [, /2; m;4z]+ 4z m 2 F [ +, /2; m +;4z] Wth the help of ths Lemma ad Gould s specal values (2, (3 we prove Proposto the form of the dettes (, 2F [ 2, /2; ;4] 2 F [ 2, /2; ;4]+ 8 2 F [ 2 +, /2; +;4]3+0 2F [ 2 +, /2; ;4] 2 F [ 2 +, /2; ;4]+ +4 (5 4(2 2F [ 2 +2, /2; +;4] There are may compaos to the the par of dettes Proposto We prove oe example Corollary 3 For postve tegers holds ( 3 2 ( ( 3 C , + ( ( 3 ( C 2 Proof The specal values (4 show that (2+ 2F [ 2, /2; +;4] F [ 2 +, /2; +2;4] By Lemma F [ 2, /2; +;4] 2 ( /2 j 2 2j ( 2 j j0 2 2 (2j 2!( 2 j j! j thus ( j0 ( j j + ( ( 2j j (+ j 3 2 j j!(j!(+ j Ths detty together wth ( ( ( C (see Proposto mply that ( ( ( ( ( ( The Lattce Path Approach Let D (,m y(x c/g be the umber of, -paths strctly above y (x c /g from the org to (, m, where c s ay o-egatve teger The utagled detours (Table 5 show that d (, m D (,m y(x /2 Because classcal lattce path eumerato usually cosders paths above les of teger slope, we use two bjectos o, -paths to fd the electroc joural of combatorcs 9 (2002, #R33 8

9 D (,m y(x c/g Gog backwards, from the ed to (0, 0, through ay path strctly above y (x c /g to (, d +( c /g yelds a uque path strctly below y d + x/g, aga to (, d +( c /g, ad vce versa (ote that d +( c /g must be a teger The secod bjecto reflects ths ew path at the dagoal y x oto a path strctly above y g (x d, edg at (d +( c /g, D (,m y(x /2 D(,m (# paths to (6, 5 m y2x 5 (# paths to (5, 6 m Table 7: Two dfferet lattce path problems arrvg at the same umber The result s a path above a le wth postve teger slope g, ad we have establshed that D (,d+( c/g y(x c/g D (d+( c/g, yg(x d The umber of paths above a le wth postve teger slope s well kow to be d D (m, yg(x d ( ( + g ( d g (m d m + g ( d g ( d m ( m + m ( ( + g ( d g (m d m + g ( d m g ( d m d + for m g (m d (for refereces see Koroljuk [4] or Mohaty [6] The result also follows from (4, wth u g, v gd, ads (u + v ( +g( d for 0 d, ad s (u + v 0for>d I hoor of Koroljuk s work we wll also gve a combatoral proof of the formula for D (,m y(x c/g usg hs method at the ed of ths paper, Proposto 4 We just saw that D (m, y2(x m+( /2 D(,m y(x /2 d (, m, ad therefore m ( /2 ( ( 3 2m + 3(m d (, m 2(m + m ( ( /2 m + ( m 3 + C m m ( ( /2 m + ( m ( C 2 (7 the electroc joural of combatorcs 9 (2002, #R33 9

10 for m ( /2 (remember that C ( 3 / (2 +, the Tr-Catala umbers Specal values, lke 2m, wll cofrm some of the dettes Corollary 3 The last of the three expasos of d (, m (7 cofrms that d (, m s a polyomal m of degree Hece for weobtad (, ( /2 ( ( ( 3 ( ( /2 ( 2 + ( + (8 We ca fsh ths lattce path proof of d (, ( C by showg that the latter sum equals ( 2 2 (see [3, (369], but t s also structve to compare the geeratg fuctos Frst otce that 0 t ( t 3 ad C t 2 + ( 3 Thus t ( d (, ( +3 3 t 2 F [ 3, 2 3 ; 3 2 ; 27 4 t C t 2 F 2+ ] 3s( arcs ( 27t/2 3 27t/2 ( + t 3 [ 3, 2 3 ; 3 2 ; U 2 ] 3s( arcs U 3 ( t U C t2 ( t 3 where U 3 3t 2 2 / ( t 3 We ca elmate the /3 frot of the arcs by fdg a θ such that s 3θ U Defe θ such that s θ U ( t ( 4t / (6t Applyg the trple-agle formula s 3θ 3sθ 4s 3 θ shows that deed s 3θ U Hece t ( d (, 3s(θ ( t U 2 + 4t 0 3( t ( 4t 4t ( t6t 2t C t, the well-kow geeratg fucto of the Catala umbers (I thak the referee for shorteg the dervato by meas of the trple-agle formula We wll have aother look at ths geeratg fucto detty Secto 53 5 Some Remarks o Paths above a Fractoal Slope There are more types of kow umber sequeces occurrg the Catala traffc scheme tha just the Catala umbers, ad there are other lattce path problems restrcted by a boudary le of slope /2 where the alteratg Catala umbers play a role A short dscusso follows the ext two subsectos Fally, we fsh wth a glmpse at the eumerato of lattce paths above a le of fractoal slope the electroc joural of combatorcs 9 (2002, #R33 0

11 5 Other types of umbers geerated by Catala traffc m C 3 2 gate C 2 3 gate Ξ 2 C gate Ξ C 0 gate Ξ M 0 M M 22 M 3 4 M 49 M 5 2 t (, m Table 8: Motzk ad Tr-Catala umbers the Catala traffc table Formulas (7 ad (7 are smple expressos for the umber D (,m y(x /2 of, -lattce paths from the org to (, m stayg strctly above the boudary y (x /2, ( m + D (,m y(x /2 d (, m j0 ( ( /2 m + +( m C ( j C j ( m + j ( m 2 for all 2m+, ad also for the polyomal exteso to other values of m (aga, we use the abbrevato C ( 3 / (2 + It s ow easy to prove that the Motzk umbers M ( k ( + + k0 k k occur as t (, the Catala traffc Table 2, ad equvaletly as ( d ( +2, 2 Table 6 To check ths, ote that 0 d ( 2 t ( 3( t +2t 3t 2 /2, a specal case of the geeratg fucto 0 d (x t ( 3 t +3t ( t x 3/2 /2 (see (8 Hece ( d ( 2 t 2 t ( 2 t ( 3t(+t /2, 2 the geeratg fucto of the Motzk umbers We ca apply the expaso d (x d ( (u + v x u v +x u v x u v (see (4 to express the umbers d (, m terms of the Motzk umbers, ( ++m d (, m d ( 2 ( ( ++m + m 2 ( +m 2 + d +2 ( 2 2 m + ( + m 2 ( +m + ( M (20 m + 2 (9 the electroc joural of combatorcs 9 (2002, #R33

12 The umbers at the gates Table 2, or d (2m, m Table 6, are the Tr-Catala umbers ; ths follows from (9 ad the frst detty Corollary 3, C m d (2m, m ( 2m 3m m j0 2m + ( 3m m ( ( j 3m C j 2m j C m ( ( 3m 3m 2 m 2m + I the followg table we summarze the verse relatoshps ( the sese of Rorda [9, Chpt 2] betwee the three types of umbers, Catala, Motzk, ad Tr-Catala, usg (9, (20, ad Corollary 3 /2 C ( + d + ( M ( d ( +2, 2 C d (2, + (+/2 ( ( + C 2 ( + C ( M ( 3 + ( M ( ( + C + 2 ( ( 2 C 3 2 Because of the boudary y (x /2 we have the tal values d (2m +,m d (2m, m 0 for all m > 0 (3, mplyg that d (2m, m d (2m,m d (2m 2,m Each of the three equal terms we ca expad wth formula (9, as we dd for d (2m, m above Expadg the dfferece d (2m, m d (2m,m0showsthat ( 2m 3m ( 3m ( j C j m 2m j j0 recoverg the secod equato Proposto The frst follows aalogously I Proposto 4 we wll apply a stadard combatoral argumet to prove that d (2m, m C m, wthout the help of Catala umbers or ay polyomal extesos However, I dd ot fd a drect combatoral proof showg that the Catala umbers must occur the traffc scheme; all the argumets above are based o polyomal extesos of D (,m y(x /2 to values that ca ot be terpreted as a umber of lattce paths 52 Other lattce path problems geeratg alteratg Catala umbers There are closely related lattce path problems where the alteratg Catala umbers play a smlar role A trval varato s the umber D (,m yc+(x /2 of all, -paths from (0, 0 to (, m stayg strctly above y c +(x /2, for some postve teger c I the electroc joural of combatorcs 9 (2002, #R33 2

13 d (m m f (m Table 9: The polyomal extesos d (m ofd(, m, ad f (m off(, m ths case all paths beg wth upwards steps, ad therefore D (,m yc+(x /2 D(,m c y(x /2 d (, m c Hece D (,c yc+(x /2 d (, ( C I Table 9 the umbers f (, m :D (,m yx/2 of all, -paths from (0, 0 to (, m stayg strctly above y x/2 (except at the org look very dfferet from d (, m - except for the alteratg Catala umbers! We wll show below that f ( +, ( C for postve (ote that f (, 00adf (0, The backwards dfferece recurso for those paths has tal values f (2m, m f (2m +,m 0, except for f (0, 0 We apply the expaso f (x f ( (u + v x u v +x u v x u v (see (4 to the polyomal exteso f (x off (, m, wth u adv f (x ( ( +x +x f ( + ( +x f ( Substtutg the tal values f (2m, m δ 0,m ad f (2m +,m0showsthat ( 3m 2m 2m 2 f ( 2 ( ( 2m+ 3m 3m ad 2m 2m + 2 ( f ( 3m 2m + Therefore we ca determe f ( as the uque solutos of the system of equatos ( ( ( ( 2m 3m 3m 3m 3m ( 3m + f + (, 2m 2m 2m 2m 2m ( ( ( ( m + 3m 3m 3m 3m 2m ( 3m + f + ( m 2m + 2m 2m + 2m 2m From equato (0 Proposto ad the secod equato Corollary 3 follows the electroc joural of combatorcs 9 (2002, #R33 3

14 f + ( ( C for postve Hece ( +x ( +x f (x + ad f (x t ( 2 ( t x 3t + (3t +( t 0 The specal cases f (2m 3,mf (2m 2,mf (2m,m ( 3m 2 2m + 2m 2 ( ( C 3m 2 2m 2 ca be summed, because by (9 2m 2 ( ( ( 3m 2 3m 2 3m 2 ( C 2m 2 2m 2m 2 thus f (2m 3,mf (2m 2,m ( ( ( 3m 2 3m 2 3m 2 f (2m,m2 (2 2m 2m 2 2m 2m 2 ( The umbers 3m 2 ( 2m 2m 2 2 3m 3 2m / (3m, m, are sequece A00603 the O- Le Ecyclopeda of Iteger Sequeces They occur the eumerato of ocrossg trees o a crcle [8] Aga, a trval varato s the umber D (,m yc+x/2 f (, m c ofall, -paths from (0, 0 to (, m stayg strctly above y c + x/2, for some postve teger c 53 Slopes that are half-tegers Applyg hs probablstc method, Ira Gessel [] foud geeratg fuctos for eumeratg paths below half-teger slopes Let λ be a oegatve, s a postve teger, ( ad ( deote by g m the umber of, -paths strctly below y 2λ+ x + s /2 to 2 2m, s + λ + 2 2m, ad by hm the umber of paths strctly below y 2λ+ x + s 2 to ( 2m +,s + ( λ (2m + Let q (t be the power seres defed by t ( q (t q (t λ+/2 Gessel showed by Lagrage verso that ( q (t a a (λ +3/2 m + a t m (λ +3/2 m + a m m0 For example, let a, λ 3/2, ad we wll obta the geeratg fucto of the Tr-Catala umbers, f we oly solve t ( q (t q (t 2 for q (t Ths s ot a smple expresso; however, we observed Secto 4 that the substtuto ( t τ 2 ( τ 3 eormously smplfes the formal power seres, gvg us q (τ 2 4τ ( τ Gessel s probablstc method shows that m0 g m t 2m q (t s+/2 + q ( t s+/2 q (t /2 + q ( t /2 ad h m t 2m+ q (t s + q ( t s q (t /2 + q ( t /2 the electroc joural of combatorcs 9 (2002, #R33 4 m0

15 As a example, Gessel evaluated the case s, ad foud ( (2λ +3m + λ +2 h m (2λ +3m + λ +2 2m + ( (λ +3/2 (2m ++/2, (λ +3/2 (2m ++/2 2m + (22 the umber of paths strctly below y 2λ+ x +to ( 2m +, + ( λ (2m + Traversg backwards through a path, from the ed to the org, shows that every path form (0, 0 to ( 2m +,s + 2λ+ (2m + strctly below y s + 2λ+ x (except at the ed pot, correspods to a path from (0, 0 to ( 2m +,s + 2λ+ (2m +, 2 2 strctly above y 2λ+ x If λ 0 ad s, ths becomes a path from (0, 0 to 2 (2m +,m+, strctly above y x/2, ad h m ( 3m +2 3m +2 2m + ( 3m +2 3m +2 2m + accordg to (22 as well as (2 the precedg subsecto 54 Slopes that are ut fractos ( 3m + f (2m +,m+ 2m + 2m Gessel s approach works well for half-teger slopes For ut fractos, slopes of the form /g, we ca fd the umber of paths from the bjectos we dscussed at the start of Secto 4, showg that D (,m y(x d/g D(m, yg(x m+( d/g The followg elemetary approach mrrors Koroljuk s method for postve teger slopes [4] Ξ Ξ 0 Ξ D (,m y(x 3/ Table 0: { }-paths strctly above 4y + 3 x Proposto 4 Suppose g ad c are postve tegers, ad D (k y(x c/g s the set of, - paths from the org to (gk + c, k stayg strctly above the boudary gy + c x, except at the ed pot (gk + c, k The D (gk+c,k y(x c/g c (g +k + c ( (g +k + c k, the electroc joural of combatorcs 9 (2002, #R33 5

16 a geeralzed Catala umber Furthermore, ( D (,m + m y(x c/g ( c/g ( ( (g + + c + m (g + c m c (g + + c Proof Defe D (k D y(x c/g (gk+c,k y(x c/g Ay (urestrcted path from (0, 0 to (gk + c, k ca be decomposed to a head pece that reaches the le gy x c for the frst tme at (gj + c, j, say, ad a tal from (gj + c, j to(gk + c, k wthout ay restrctos Hece ( gk + c + k k k D ( y(x c/g ( g (k +k k Applyg the expaso s k (gk + c k s ( c (g (g+(k +c (g+(k +c k to the polyomals s (x ( g+x (see (4 shows the well-kow detty ( (g +k + c k ( ( g + c (g +(k +c k (g +(k +c k Hece D ( y(x c/g ( c (g+k+c (g+k+c k We apply the dea of head ad tal peces aga, but ow to the path from (0, 0 to (, m thatreachatleastoepotothelegy+c such paths Ths proves the formula for x There are ( c/g D ( y(x c/g ( g c+m m D (,m y(x c/g The geeralzed Catala umbers occur the eumerato of g + -ary trees ad the g + -tes ball problem [5] For g 2adc the Proposto verfes the Tr-Catala umbers at the gates Table 2, because Refereces D (gk+c,k y(x c/g D (gk+c,k y(x c/g [] Gessel, I (986 A probablstc method for lattce path eumerato J Statst Pla Iferece 4, [2] Gould, HW (973 Improved evaluato of the fte hypergeometrc seres F (, /2; j +Proc West Vrga Acad Sc 45, [3] Gould, HW (972 Combatoral Idettes Morgatow, Va [4] Koroljuk, VS (955 O the dscrepacy of emprc dstrbutos for the case of two depedet samples Izvestya Akad Nauk SSSR Ser Mat 9, 8-96 (traslated IMS & AMS 4 ( [5] Merl, D G, Sprugol, R, ad Verr, M C (2002 The tes ball problem To appear J Comb Th A the electroc joural of combatorcs 9 (2002, #R33 6

17 [6] Mohaty, SG (979 Lattce Path Coutg ad Applcatos Academc Press, New York [7] Nederhause, H (980 Lear recurreces uder sde codtos Europea J Comb, [8] Noy, M (998 Eumerato of ocrossg trees o a crcle, Dscr Math 80, [9] Rorda, J (968 Combatoral Idettes, Wley, New York [0] Staley, RP (999 Eumeratve Combatorcs, Vol II, Cambrdge Uversty Press, Cambrdge, UK the electroc joural of combatorcs 9 (2002, #R33 7