# Combinatorial Interpretation of Raney Numbers and Tree Enumerations

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2 C. H. Pah, M. R. Wahiddi wee itoduced ealie tha Catala ubes [9], the Catala ubes ae oe popula ad widely used tha the Fuss-Catala ubes (see [10] [11] fo details). Due to its self siila stuctue, the applicatios of Catala ubes could be foud i ay physical pobles, e.g., lattice odel [12], tee eueatio etwo [13], ad Hael atices i codig theoy [14]. A tee is a coected gaph with o cycles ad fo which oly oe shotest path exists fo oe ode to aothe. Tee eueatio is a ipotat tool to study etwo. These etwos always gow i a powe-law behavio which is ofte foud i social etwo, subway syste [15], etc. R i the fo of a o-liea ecusio ad the we povide a cobiatoial itepetatio of Raey ubes. Usig this cobiatoial itepetatio, we solve seveal tee eueatio coutig pobles i which we ecove the well-ow Fuss-Catala ubes [16], Catala tiagles [17], ad othe less ow ubes. Motivated by the coectio betwee Raey ubes ad Catala tiagles, a geealizatio of Catala tiagles is poposed ad we pove soe of thei popeties. Cosequetly these foulas geealize the popeties of Catala tiagles. Fo the exact solutio of these tee eueatio pobles, we ae able to fid a shap uppe boud of the ube of each tee eueatio poble. The uppe boud is ipotat i the cotou ethod fo lattice odels ad liit of the ado gaph. I this pape, we itoduce Raey ubes (, ) 2. Raey Nubes Let C ( ) be the ube of a -ay tees with labeled vetices (Figue 1), whee 1 C ( ) =, 2, 1. ( 1) + 1 The Raey ubes ae defied as follows: R, = C i C i C i, C 0 = 1, > 0 (1) whee i, i,, i { 0} ( ) ( 1) ( 2) ( ) ( ) i1+ i2+ + i = 1 2. Theefoe, the cobiatoial itepetatio is as follows: copies of -ay tee with total ube of vetices. Next, we let u ( x ) be the geeatig fuctio fo C ( ), i.e., 2 u x = 1+ C 1 x+ C 2 x + + C x +. ( ) ( ) ( ) ( ) The, the geeatig fuctio of R (, ) is u ( ) [18]. Lea 1. Let u ( ) x be the geeatig fuctio of the Raey ubes. The, ( ( )) + x u x =. + x ad the Raey ubes satisfy the followig foula Iediately, we obtai the followig theoe. Theoe 1. The bioial fos of the Raey ubes ae give by + R (, ) =. + Fo theoe (1), it is ot difficult to deduce soe of the popeties of Raey ubes. Coollay 1. Fo itege > 1, we have ( ) R 0, = 1; (4) (2) (3) Figue 1. A biay tee with 3 odes, whee the botto vetex is the oot. 2

3 C. H. Pah, M. R. Wahiddi i R ( i, i) = ; i 1 R ( 1, ) = R (,1 ) = ; ( 1) + 1 R + 1, =. 1 ( ) Coollay 2. We ca wite C ( ) i a oliea ecusio as: C ( ) C ( i1) C ( i2) C ( i) C ( ) whee i, i,, i { 0} + 1 =, 0 = 1 (8) i1+ i2+ + i = 1 2. We ecove the foula by joiig the copies of -ay tee with vetices which is also equivalet to a -ay tee with vetices ad a additioal oot (see Figue 2): Usig bioial fo of R (, ) ( ) = ( + ) R, R 1,1., oe ca obtai the followig esult. Coollay 3. Fo a fixed itege > 1, ad > 1, whee ( ) ( 1, 1 ) (, ) (, 1) R + = R R (9) R, = 0 if <. Fo = 2, we ecove the idetity of a geealized Ballot ubes: 3. A Hoogeeous -Ay Tee ( ) ( ) ( ) R 1, + 1 = R, R, 1. (10) Ulie the usual -ay tee, we defie a hoogeeous -ay tee as a gaph with o cycles, i which each vetex eaates + 1 edges (see Figue 3 fo = 4). We fix a vetex aely z as the oot. Ulie the odiay oot i a -ay tee, this oot has + 1 successos while othe vetices have ube of successos. Ay vetex could be chose to be the oot sice the gaph is hoogeous. Fo a give vetices, we ay fid how ay coected sub-tee ooted at z. This ube is defied as D ( ). Theoe 2. Fo 1 C as: >, we ca wite D ( ) i a oliea ecusio of ( ) ( ) ( ) ( ) D = C C fo > 0. (11) = 1 Poof. We decopose the poble by fidig out the ube of -ay tee of ube of oe copy of C, ad aothe copy of -ay tee with C. -ay tee with vetices, i.e., ( ) (5) (6) (7) vetices, i.e. ( ) Figue 2. Joiig 4 ooted Cayley tee of ode 4 whee = 4 ad = 4. 3

4 C. H. Pah, M. R. Wahiddi Figue 3. A hoogeous gaph whee each vetex is coected to exactly 5 eighbous. x, its age should be fo 1 to. Total D ( ) 0 Sice the foe C ust always iclude of all C ( ) C ( ) usig the additio ad ultiplicatio piciples. Usig Equatio (8), we ewite the foula above as ( ) ( 1) ( 2) ( + 1) = 1 is just the su D = C C C fo > 0. (12) This foula ca also be obtaied usig + 1 copies of -ay tee togethe with 1 vetices ad oe cete. We the fid the bioial fo of D ( ). Coollay 4. Fo 2, D ( ) is expessed i bioial fo as: + 1 D ( ) = R ( 1, + 1) = > 0. (13) ( 1) Thus, D ( ) = ad D ( ) = as i [19] ad [20], espectively. Fo = 4, D ( ) coicides with oe fo of Raey ubes as etioed above, i.e., R4 (,5). The ubes D ( ) a lot of ew sequeces. Fo exaple, the sequece of R ( ), i.e., is ot foud i the OEIS database [6]. Fo theoe (2), oe ca get 5,6 1,6, 45,380,3450,32886,324632, , , , ( ) = ( ) ( ) ( ) D C C C = 0. 4 geeate This foula ca be also obtaied easily by a diffeet way: 1) Cout the ube of tees by joiig the 2 copies of ay tee, with total ube of vetices. 2) Subtact those tees eueate fo y but does t cotai z, that is, exactly the ube C ( ). Let the geeatig fuctio of D ( ) be w ( x ). The we have the followig esult. Coollay 5. Fo > 1 ad 0 w x is whee ( ), the geeatig fuctio, ( ) 2 w ( x) u ( x) u ( x) u x is the geeatig fuctio of C ( ). Coollay 6. Fo > 1, = (14) 4

5 C. H. Pah, M. R. Wahiddi o ( ) ( ) = ( 1, + 1) C C R (15) = 1 = 1 ( ) ( ) = ( + ) R,1 R,1 R 1, 1. (16) Usig the bioial iequality i [21] ad the bioial fos of ( ) equality ca be easily poved. Coollay 7. Fo 2 >, whee b = 1 1 ad > 0. ( ) ( ) ( b ) 32 C ad D ( ), the followig i- C D < (17) Fo sufficietly lage, a siple fo is poduced as well, i.e., C ( ) D ( ) cojectued i a weae fo i [20], i.e., C ( ) D ( ) 4. Catala Tiagle A Catala tiagle B(, ) is defied as follows [9]: The Catala tiagle satisfies [9]: ( e) <. ( e) 1 if = = 1; B(, ) = B( 1, 1) + 2B( 1, ) + B( 1, + 1) if 1 ; 0 othewise. ( ) 2( 1) 2( 2) 2( ) 2( ) i1 + + i = i1,, i 1 <. These esults ae 32 B, = C i C i C i, C 0 = 1 > 0 (18) Usig a popety of Catala ubes, C ( i ) C ( j ) C ( j ) fo of B(, ), i.e., whee j1, j2,, j2 { 0} ubes, i.e., B(, ) R(,2) =, whee j 1, j 2 0, we get aothe j1+ j2= i 1 ( ) ( ) ( ) ( ) B, = C j C j C j, (19) j1+ j2+ + j =. Fo Equatio (1), we iediately ecove the Catala tiagle fo the Raey = : 2 2 B(, ) = R2 (,2 ) =. We ow coside the followig poble as i [22]: Fid out the ube of all diffeet coected sub-tees of a hoogeous biay tee with ube of vetices, cotaiig the give ube of fixed vetices (whee 2 2). The coditio, 2 2, is siply the ube of vetices that coves the iial copoet cotaiig all vetices. The details of this poble ad teiologies could be foud i the oigial pape [22]. We deote the solutio to this poble as F. I this pape, we show that a solutio to the case whe the iial copoet is full, is as below: ( + ) 2 2 F = B( + 2, ) =, (20) 5

6 C. H. Pah, M. R. Wahiddi whee is the ube of give vetices ad is the ube of fixed vetices i each of the coected subtee. Now, we itepet ad elate the poble above with the cobiatoial itepetatio of the Raey ubes though the followig steps (see Figue 4): 1) Give vetices; 2) Fill up all the iteio poits, i.e., 2 ; 3) Fill up all the bouday poits, i.e., ; 4) The oly 2+ 2 vetices ae left; 5) Sice each bouday poit has 2 eighbous which is ot a iteio poit, we have 2 boxes; 6) If 2+ 2 vetices ae give, the thee ae 2 boxes of biay tee to be filled. As a esult, the solutio is ( + ) + ( + ) R2 ( 2+ 2, 2 ) = =. 2( 2+ 2) Futheoe, it is atual to defie a geealized Catala tiagle, i.e., -th Catala tiagle usig Fuss- Catala ubes istead of Catala ubes as i Equatio (19): whee ( ) ( ) ( 1) ( 2) ( ) ( ) B, = C i C i C i, C 0 = 1 > 0, (22) i1 + + i = i1,, i 1 B, = 0 if <. Fo the popety of Fuss-Catala ubes, i.e., coollay (2) C i = C j1 C j2 C j whee j1, j2,, j 0 ( ) ( ) ( ) ( ) j1+ j2+ + j = i 1, we fid aothe fo of B (, ), B ( ) C ( j ) C ( j ) C ( j ) whee j, j,, j { 0} j1+ j2+ + j =, (21), =, (23). Agai, fo Equatio (1), we iediately have B (, ) = R (, ) =. Lea 2. Soe popeties of -th Catala tiagles ae as follows: ( ) ( ) (24) B,1 = C, (25) ( ) B, = 1, (26) ( ) ( ) B, 1 = 1. (27) Figue 4. 4 bouday poits (solid cicles) coected to full iial copoet. 6

7 , oe ca show that: Lea 3. Fo > 1 ad 2 Usig the bioial fo of B (, ) If = 2, we ecove B ( 1, 1) + 2B ( 1, ) + B ( 1, + 1 ) = B2( 1, 1) + 2B2( 1, ) + B2( 1, + 1 ) = = B2(, ). C. H. Pah, M. R. Wahiddi Based o the iitial esult, lea (3), we pove the followig assetio by atheatical iductio with espect to. Theoe 3. Fo fixed, whee > 1, ad > 1, whee ( ) (28) + + B ( 1, 1 + i) =, 0 i + (29) B, = 0 if <. Poof. Assetio is tue fo = 2. Assue that it is tue fo, we coside the followig suatios: B ( 1, 1 + i) + B ( 1, + i) 0 i 0 i + + ( + 1) + + = B B i B i B 1 i 0 i + ( + 1) =. + ( + 1) ( 1, 1) + ( 1, 1 + ) + ( 1, + ) + ( 1, + ) B B i B 1 i i 1 + ( + 1) =. + ( + 1) ( 1, 1) + + ( 1, 1 + ) + ( 1, + ) + 1 B B i B 1 i + ( + 1) =. + ( + 1) ( 1, 1) + ( 1, 1 + ) + ( 1, + ) ( 1) ( 1) B ( 1, 1 + i) =. i + + Hece, the assetio is tue fo ay > 1. Coollay 8. Fo fixed, whee > 1, ad > 1, we have + i + + =. 0 i i + (30) Fo =, we have the followig siple esult: Coollay 9. Fo fixed > 1, ad > 1, B (, ) = B ( 1, 1 + i), 0 i (31) 7

8 C. H. Pah, M. R. Wahiddi B, = 0 if <. whee ( ) 5. Bioial Tasfoatio of -th Catala Tiagle Fo 2 ad, we defie a ew ube H ( ) ( ), as: ( ), ( ) ( 1) ( ) ( 1) ( ) H = C j C j C l C l. (32) j1+ + j+ l1+ + l= j1,, j 0, 1,, 1 If j1 = j2 = = j = 0, all the Fuss-Catala ubes should stat at 1, the we ecove the peviously defied -th Catala tiagle. Fo the popety of Fuss-Catala ubes, C ( i) = C ( j1) C ( j2) C ( j), whee j, j, 0, we foud aothe fo of 1 2 Fo Equatio (1), we iediately have j1+ j2+ + j = i 1 ( ), ( ) = ( 1) ( ) ( 1) ( ) H C j C j C l C l. (33) j1+ + j + l1+ + l = j1,, j, l1,, l 0 ( ) ( 0) + + H, ( ) = H, + ( ) = R(, + ) = + ad fo = 0, we ecove the sae foula fo -th Catala tiagle, ( ) ( ) ( ) H, = R (, ) =. Fo theoe (3), H, is obtaied as a esult of bioial tasfoatio of -Catala tiagles. Coollay 10. Fo fixed > 1, whee > 1, ad > 1, B, = 0 if <. whee ( ) 6. Coclusio B i H (34) ( (, ) ) + =, ( ), 0 i (35) I this pape, we have itoduced the cobiatoial itepetatio of Raey ubes to solve vaious tee eueatio coutig pobles. The uppe boud of ay + 1 ode tee eueatio is geeally foud to be ( e) (. We have also show how a ew ube H ) 32, ( ) ay be deived fo the bioial tasfoatio of -th Catala tiagles. Acowledgeets This eseach is fuded by the MOHE gat FRGS The authos ae gateful to aoyous efeee s suggestio ad ipoveet of the pesetatio of this pape. Refeeces [1] Peso, K.A. ad Zyczowsi, K. (2011) Poduct of Giibe Matices: Fuss-Catala ad Raey Distibutios. Physical Review E, 83, Aticle ID: [2] Mlotowsi, W., Peso, K.A. ad Zyczowsi, K. (2013) Desities of the Raey Distibutios. Docueta Matheatica, 18, [3] Jeuisse, R.H. (2008) Raey ad Catala. Discete Matheatics, 308, [4] Dzieiaczu, M. (2014) Eueatios of Plae Tees with Multiple Edges ad Raey Lattice Paths. Discete Mathe- 8

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