Catalan triangles and Finucan s hidden folders

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1 Notes o Number Theory ad Discrete Mathematics Prit ISSN , Olie ISSN Vol. 22, 206, No. 2, 0 6 Catala triagles ad Fiuca s hidde folders A. G. Shao,2 Faculty of Egieerig & IT, Uiversity of Techology, Sydey, NSW 2007, Australia 2 Campio College, PO Box 3052, Toogabbie East, NSW 246, Australia s: t.shao@campio.edu.au, Athoy.Shao@uts.edu.au Received: 5 August 205 Accepted: 3 February 206 Abstract: This ote is to capture ad exted some of Fiuca s ideas for further exploratio by studets. These ideas coect with several elemetary cocepts i combiatorial aalysis which led themselves to udergraduate research projects. Keywords: Catala umbers ad polyomials, Youg tableaux, Fiboacci umbers, recurrece relatios. AMS Classificatio: B39, K3, B83, 05A5. Itroductio Hery Fiuca (97 983) was Reader i Mathematics at the Uiversity of Queeslad for may years, havig studied there ad at Oxford ad Cambridge [9]. He was a ispiratioal teacher, with uorthodox isights, ad a great ecourager of youg mathematicias (which is how I met him fifty years ago). The purpose of this ote is to capture, exted ad expoud some of his ideas for further developmet by iterested researchers. I particular, his maila folder realisatio [7, 8, 9] is a excellet start for studyig the Catala umbers, Youg tableaux ad other umber triagles which ca the be exteded ito projects to ivestigate o-egative zero-retur radom walks. Fiuca s approach was very much i the hads-o spirit of the Belgia mathematicia, Eugèe Catala [5]. 2 Cotext We are asked to imagie maila folders with h folders withi them: dated I realise but quite cocrete! Each situatio produces various cofiguratios (Figures ad 2). Furthermore, suppose v spies of the folders are visible from the rear ad that u folders are uoccupied as i the tableaux of Figures 3 ad 4. Note that h = v. 0

2 The reader is ivited to try the cases for = 6 ad 7 to get a feelig for what is goig o; compare your results with Figures 5 ad 6. I the lower triagular arrays of Figures 3, 4, 5, 6, the rage of values of (u, v) is u = to, ad v = to u. We shall use Fiuca s symbol Fh to represet the umber of such cofiguratios, so that the total umber summed over all v is the Catala umber c, as we shall see. I the twoway decompositios of Figures 3, 4, 5, 6, these Catala umbers appear i the bottom right had (total) cell. Now cotiue to uravel these patters ad you should fiish up with the Catala triagle of hidde folders (Figure 7) i which the Catala umbers appear alog the backward slopig boudary diagoal. = v h 0 2 u c 2 2 Figure. Spies Figure 3. F values for = 3 =4 c represets the umber of cofiguratios; h represets the umber of iteral folders (h 0); represets the total umber of folders ( >0) h u c Figure 2. Visible spies Figure 4. F values for = 4

3 v v u u Figure 5. F values for = 6 Figure 6. F values for = Figure 7. Catala triagle of hidde folders Other features iclude row sums: Catala umbers (also o last ad secod last backward diagoals) [22], sums alog forward diagoals:,, 2, 3, 6, 0, 20, partial colum sums appear i [25]. 3 Recurrece relatios It ca be observed that the elemets, Fh, seem to be related by the partial recurrece relatio F = h F + h F (3.) h with boudary coditios a topic for further research. This recurrece relatio also appears i the cotext of coloured trees [24] which leads o to spaig trees i geeral [3]. A solutio of this recurrece relatio ca be cofirmed to be i which F h h = h is the egative biomial coefficiet defied by h ( + )( + 2)...( + h ) = h h! 2

4 with the risig factorial fuctio i the umerator [2]. For example, F 2 = = Fiuca also explored further related ideas about F(; u, v), the umber of cofiguratios at (u, v) for which he explaied the physical meaig of the recurrece relatio ( F( : u, v + ) F( ; u, )) F( ; u, v) = F( ; u, v ) + F( ; u, v) + v which has boudary coditios of the Catala umbers ad zeroes ad which has a solutio, (3.2) v v F( ; u, v) =, (3.3) u u u v which ca be readily cofirmed with ay of the previous examples. The paretheses i the recurrece relatio are there to help i explaiig the physical sigificace of the terms i relatio to the folders. Larcombe [8] has used the defiitio of the ( + ) th Catala polyomial, P(x), 0, 2 i P ( ) i ( x) = x, (3.4) i= 0 i from which we ca obtai aother type of Catala triagle from the absolute values of the coefficiets if powers of x i the polyomial: Figure 8. Catala polyomial coefficiets triagle Elemets i the th colum ad j th row are related by the partial recurrece relatio = + j j j 2 By aalogy with some of the approaches to properties of the Pascal triagle we observe that The sum alog the leadig diagoals is the sequece {,, 2, 3, 4, 6, 9, 3, 9, 28, }, which is Narayaa s sequece [6, 2], amed after a 4 th cetury Idia mathematicia ad is i practice actually defied by the third order homogeous liear recurrece relatio [5, 20] v = v + v 3, v = v2 = v3 = 3 ;

5 The sums alog the rows are the elemets of the Fiboacci sequece {,, 2, 3, 5, 8, }; The partial sums of the th colum, =, 2, 3,, are sequeces {u,j} defied by the partial recurrece relatio with uit boudaries u, j = u, j + u, j 2. 4 Cocludig commets The recurrece relatios i this ote ca be exteded alog the lies of the biomial coefficiet umbers suggested i []. Proofs eed to be developed for the partial recurrece relatio (3.2), the solutio to which i (3.3) has other combiatorial iterpretatios [3]. The Catala umbers also appear o diagoals of Youg Tableaux [0] i a ladmark paper by Leoard Carlitz ad Joh Riorda [4] which ca stimulate advaced udergraduate projects with q-geeralizatios []. The trackig of the umber relatios leads aturally ito the study of patters of lattice paths i combiatorics. The Catala umbers ca be related to Stirlig umbers [7, 22] which, i tur, ca lead back to the well-kow Pascal triagle [27]. A ecyclopaedic source of results to stimulate further related ideas may be foud i Hery Gould s bibliography [2]. It ca be see i this ote that these ideas have the attractio of ledig themselves to cocrete coutig of elemets, before tryig to geeralize: that is, guess ad test your guess a ecessary, but ot a sufficiet, part of research i mathematics ad a importat igrediet i the learig of udergraduate mathematics [23]. A further step, ot pursued here, is to try to prove that the results are geerally verifiable [26]. Ideas for further developmet of the triagles i this paper may be foud i Ver Hoggatt [4]. More particularly, Koshy has explored somewhat similar triagles associated with aother Cambridge mathematicia, aother Hery (Hery Forder), who, like Fiuca, was also a excellet teacher ad who also worked i the atipodes (albeit New Zealad) [6]! Refereces [] Adrews, G. E. (987) Catala umbers, q-catala ad hypergeometric series. Joural of Combiatorial Theory, Series A. 44, [2] Barbero, S., & Cerrutti, U. (200) Catala Momets. Cogressus Numeratium. 20, [3] Bejami, A. T., & Yerger, C. R. (2006) Combiatorial Idetities of Spaig Tree Idetities. Bulleti of the Istitute of Combiatorics ad Its Applicatios. 47, [4] Carlitz, L., & Riorda, J. (964) Two Elemet Lattice Permutatio Numbers ad their q- geeralizatio. Duke Mathematical Joural. 3, [5] Catala, E. (885) Quelques théorèmes empiriques. Mémoires de la Société Royale des Scieces de Liège. 2,

6 [6] Che, W. Y. C., Wag, L. X. W., Yag, A. L. B. (200) Schur Positivity ad the q-logcovexity of the Narayaa Polyomials. Joural of Algebraic Combiatorics. 32, [7] Fiuca, H. M. (977) Some Elemetary Aspects of the Catala Numbers. I Louis R. A. Casse, Walter D. Wallis. (eds). Combiatorial Mathematics IV. Berli, Spriger-Verlag, [8] Fiuca, H. M. (98) Fragmetig the Erdelyi-Etherigto-Sads Number. Departmet of Mathematics, Uiversity of Queeslad. Colloquium. 0 th August. [9] Fiuca, H. M. (982) Some Decompositios of Geeralised Catala Numbers. I Ae Pefold Street, Elizabeth J. Billigto, Sheila Oates-Williams. (eds). Combiatorial Mathematics IX. Berli, Spriger-Verlag, [0] Fulto, W. (997) Youg Tableaux, With Applicatios to Represetatio Theory ad Geometry. Cambridge, Cambridge Uiversity Press. [] Glaister, P., & Glaister, E. M. (204) Alteratig Sums of Biomial Coefficiets with Uit Fractio Arithmetic Sequece Coefficiets. Iteratioal Joural of Mathematical Educatio i Sciece ad Techology. 5, [2] Gould, H. W. (972) Combiatorial Idetities, Secod editio. Morgatow, WV, Published by the author. [3] Gould, H. W. (978) Bell ad Catala Numbers, Research Bibliography of Two Special Number Sequeces. Revised Editio. Morgatow, WV, Combiatorial Research Istitute. [4] Hoggatt, V. E. Jr. (968) A New Agle o Pascal s Triagle. The Fiboacci Quarterly. 6, [5] Jarde, D. (966) Recurrig Sequeces. Jerusalem, Riveo Lematematika. [6] Koshy, T. (204) Lobb Numbers ad Forder s Catala Triagle. Bulleti of the Istitute of Combiatorics ad Its Applicatios. 7, [7] Lag, W. (2000) O Geeralizatios of the Stirlig Number Triagles. Joural of Iteger Sequeces. 3, Article [8] Larcombe, P. J., O Neill, S. T., & Feessey, E. J. (204) O certai series expasios of the sie fuctio, Catala umbers ad covergece. The Fiboacci Quarterly. 52, [9] Lipto, S. (985) Obituary, Hery Maurice Fiuca Joural of the Australia Mathematical Society. (Series A). 38, 8. [20] Masour, T., & Shattuck, M. (203) Polyomials whose coefficiets are geeralized Triboacci umbers. Applied Mathematics ad Computatio. 29, [2] Narayaa, T. V. (955) Sur les treillis formée par les partitios d u etire etleurs applicatios à la théorie des probabilités. Comptes redus hebdomadaires des séaces de l Académie des Scieces. (Paris). 240,

7 [22] Rogers, D. G. (978) Pascal Triagles, Catala Numbers ad Reewal Arrays. Discrete Mathematics. 22, [23] Shao, A. G. (99) Shrewd guessig i problem-solvig, Iteratioal Joural of Mathematical Educatio i Sciece ad Techology. 22, [24] Shao, A. G., Turer, J. C., & Ataassov, K. T. (99) A Geeralized Tableau Associated with Colored Covolutio Trees. Discrete Mathematics. 92, [25] Sloae, N. J. A., & Plouffe, S. (995) The Ecyclopedia of Iteger Sequeces. Sa Diego, CA, Academic Press. [oeis. org]. [26] Turer, J. C., & Shao, A. G. (993) O a ihomogeeous, o-liear, secod-order recurrece relatio, Iteratioal Joural of Mathematical Educatio i Sciece ad Techology, 24, [27] Wright, T. (995) Pascal s Triagle Gets Its Gees from Stirlig Numbers of the First Kid. College Mathematics Joural. 26,