Combinatorial Proofs of Fibonomial Identities

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1 Clareot Colleges Clareot All HMC aculty Publicatios ad Research HMC aculty Scholarship Cobiatorial Proofs of ibooial Idetities Arthur Bejai Harvey Mudd College Elizabeth Reilad Recoeded Citatio Cobiatorial Proofs of ibooial Idetities (with Elizabeth Reilad*) The iboacci Quarterly, Vol. 52, Nuber 5, pp. 28-4, Deceber This Article is brought to you for free ad ope access by the HMC aculty Scholarship at Clareot. It has bee accepted for iclusio i All HMC aculty Publicatios ad Research by a authorized adiistrator of Clareot. or ore iforatio, please cotact scholarship@cuc.clareot.edu.

2 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES ARTHR T. BENJAMIN AND ELIZABETH REILAND Abstract. ibooial coefficiets are defied lie bioial coefficiets, with itegers replaced by their respective iboacci ubers. or exaple, Rearably, is always a iteger. I 2010, Bruce Saga ad Carla Savage derived two very ice cobiatorial iterpretatios of ibooial coefficiets i ters of tiligs created by lattice paths. We believe that these iterpretatios should lead to cobiatorial proofs of ibooial idetities. We provide a list of siple looig idetities that are still i eed of cobiatorial proof. 1. Itroductio What do you get whe you cross iboacci ubers with bioial coefficiets? ibooial coefficiets, of course! ibooial coefficiets are defied lie bioial coefficiets, with itegers replaced by their respective iboacci ubers. Specifically, for 1, or exaple, ,5. ibooial coefficiets reseble bioial coefficiets i ay ways. Aalogous to the Pascal Triagle boudary coditios 1 ad 1, we have 1 ad 1. We also defie 0 1. Sice +( ) , Pascal s recurrece the followig aalog has Idetity 1. or 2, As a iediate corollary, it follows that for all 1, is a iteger. Iterestig iteger quatities usually have cobiatorial iterpretatios. or exaple, the bioial coefficiet a couts lattice paths fro (0, 0) to (a, b) (sice such a path taes a + b steps, a of which are horizotal steps ad the reaiig b steps are vertical). As described i [1] ad elsewhere, the iboacci uber +1 couts the ways to tile a strip of legth with squares (of legth 1) ad doios (of legth 2). As we ll soo discuss, ibooial coefficiets cout, appropriately eough, tiligs of lattice paths! 2. Cobiatorial Iterpretatios I 2010 [9], Bruce Saga ad Carla Savage provided two elegat coutig probles that are euerated by ibooial coefficiets. The first proble couts restricted liear tiligs ad the secod proble couts urestricted bracelet tiligs as described i the ext two theores. 1

3 2 ARTHR T. BENJAMIN AND ELIZABETH REILAND Theore 2. or a, b 1, a couts the ways to draw a lattice path fro (0, 0) to (a, b), the tile each row above the lattice path with squares ad doios, the tile each colu below the lattice path with squares ad doios, with the restrictio that the colu tiligs are ot allowed to start with a square. Let s use the above theore to see what is coutig. There are 20 lattice paths fro (0, 0) to (, ) ad each lattice path creates a iteger partitio ( 1, 2, )where 1 2 0, where i is the legth of row i. Below the path the colus for a copleetary partitio ( 1, 2, )where or exaple, the lattice path below has horizotal partitio (, 1, 1) ad vertical partitio (0, 2, 2). The first row ca be tiled 4 ways (aely sss or sd or ds where s deotes a square ad d deotes a doio). The ext rows each have oe tilig. The colus, of legth 0, 2 ad 2 ca oly be tiled i 1 way with the epty tilig, followed by tiligs d ad d sice the vertical tiligs are ot allowed to begi with a square. or aother exaple, the lattice path associated with partitio (, 2, 2) (with copleetary vertical partitio (0, 0, 2)) ca be tiled 12 ways. These lattice paths are show below. (,) (,) ways ways 1way 1way 1 1 w w ay ay 2 ways 2 ways 1 w ay (0,0) (0,0) igure 1. The rows of the lattice path (, 1, 1) ca be tiled ways. The colus below the lattice path, with vertical partitio (0, 2, 2) ca be tiled 1 way sice those tiligs ay ot start with squares. This lattice path cotributes tiligs to. The lattice path (, 2, 2) cotributes 12 tiligs to The lattice path associated with (, 1, 0) has o legal tiligs sice its vertical partitio is (1, 2, 2) ad there are o legal tiligs of the first colu sice it has legth 1. There are 10 lattice paths that yield at least oe valid tilig. Specifically, the paths associated with horizotal partitios (,, ), (, 2, 2), (, 1, 1), (, 0, 0), (2, 2, 2), (2, 1, 1), (2, 0, 0), (1, 1, 1), (1, 0, 0), (0, 0, 0) cotribute, respectively, tiligs to. More geerally, for the ibooial coefficiet a,wesuoverthe a lattice paths fro (0, 0) to (a, b) which correspods to a iteger partitio ( 1, 2,..., b )wherea 1 2 b 0, ad has a correspodig vertical partitio ( 1, 2,..., a )where a b. Recallig that 0 0 ad 1 1, this lattice path cotributes b a 1 tiligs to a. The secod cobiatorial iterpretatio of ibooial coefficiets utilizes circular tiligs, or bracelets. A bracelet tilig is just lie a liear tilig usig squares ad doios, but bracelets.

4 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES also allow a doio to cover the first ad last cell of the tilig. As show i [1], for 1, the Lucas uber L couts bracelet tiligs of legth. or exaple, there are L 4 tiligs of legth, aely sss, sd, ds ad d s where d deotes a doio that covers the first ad last cell. Note that L 2 couts ss, d ad d where the d tilig is a sigle doio that starts at cell 2 ad eds o cell 1. or cobiatorial coveiece, we say there are L 0 2epty tiligs. The ext cobiatorial iterpretatio of Saga ad Savage has the advatage that there is o restrictio o the vertical tiligs. Theore. or a, b 1, 2 a couts the ways to draw a lattice path fro (0, 0) to (a, b), the assig a bracelet to each row above the lattice path ad to each colu below the lattice path. Specifically, the lattice path fro (0, 0) to (a, b) that geerates the partitio ( 1, 2,..., b ) above the path ad the partitio ( 1, 2,..., a ) below the path cotributes L 1 L 2 L b L 1 L 2 L a bracelet tiligs to 2 a. Note that each epty bracelet cotributes a factor of 2 to this product. or exaple, the lattice path fro (0, 0) to (, ) with partitio (, 1, 1) above the path ad (0, 2, 2) below the path cotributes L L 1 L 1 L 0 L 2 L 2 72 bracelet tiligs euerated by (,) L 4 ways 1way 1wayways ways (0,0) L 0 2 ways igure 2. The rows above the lattice path ca be tiled with bracelets i 4 ways ad the colus below the path ca be tiled with bracelets i L 0 L 2 L ways. This cotributes 72 bracelet tiligs to I their paper, Saga ad Savage exted their iterpretatio to hadle Lucas sequeces, defied by 0 0, 1 1 ad for 2, a 1 + b 2. Here +1 euerates the total weight of all tiligs of legth where the weight of a tilig with i squares ad j doios is a i b j. (Alteratively, if a ad b are positive itegers, +1 couts colored tiligs of legth where there are a colors for squares ad b colors for doios.) Liewise the uber of weighted bracelets of legth is give by V av 1 + bv 2 with iitial coditios V 0 2 ad V 1 a (so the epty bracelet has a weight of 2). This leads to a cobiatorial iterpretatio of Lucasoial coefficiets, defied lie the ibooial coefficiets. or exaple, Both of the previous cobiatorial iterpretatios wor exactly as before, usig weighted (or colored) tiligs of lattice paths.

5 4 ARTHR T. BENJAMIN AND ELIZABETH REILAND. Cobiatorial Proofs Now that we ow what they are coutig, we should be able to provide cobiatorial proofs of ibooial coefficiet idetities. or exaple, Idetity 1 ca be rewritte as follows. Idetity 4. or, 1, Cobiatorial Proof: The left side couts tiligs of lattice paths fro (0, 0) to (, ). How ay of these tiled lattice paths ed with a vertical step? As show below, i all of these lattice paths, the first row has legth ad ca be tiled +1 ways. The rest depeds o the lattice path fro (0, 0) to (, 1). Suig over all possible lattice paths fro (0, 0) to (, 1) there are + 1 tiled lattice paths for the rest of the lattice. Hece the uber of tiled lattice paths edig i a vertical step is ways + 1 ways (, ) (, 1) (0, 0) igure. There are tiled lattice paths that ed with a vertical step. How ay tiled lattice paths ed with a horizotal step? I all such paths, the last colu has legth ad ca be tiled 1 ways (begiig with a doio). Suig over all lattice paths fro (0, 0) to ( 1,) there are tiled lattice paths for the rest of the lattice. Hece the uber of tiled lattice paths edig i a horizotal step, as illustrated below, is (0, 0) tiled lattice paths that ed with a hori- igure 4. There are zotal step. ( 1,) (, ) ways 1 ways doio Cobiig the two previous cases, the total uber of tiled lattice paths fro (0, 0) to (, ) is

6 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES 5 Replacig liear tiligs with bracelets ad reovig the iitial doio restrictio for vertical tiligs, we ca apply the sae logic as before to get L L. 1 Dividig both sides by gives us Idetity 5. or, 1, L + L. 1 I full disclosure, Idetities 4 ad 5 are used by Saga ad Savage to prove their cobiatorial iterpretatios, so it is o surprise that these idetities would have easy cobiatorial proofs. The sae is true for the weighted (or colorized) versio of these idetities for Lucasoial coefficiets. Idetity. or, 1, + Idetity 7. or, 1, V + V. 1 By cosiderig the uber of vertical steps that a lattice path eds with, Reilad [8] proved Idetity 8. or, 1, + j 1+ j +1 j 1 1 j0 Cobiatorial Proof: We cout the tiled lattice paths fro (0, 0) to (, ) by cosiderig the uber j of vertical steps at the ed of the path, where 0 j. Such a tilig begis with j full rows, which ca be tiled j +1 ways. Sice the lattice path ust have a horizotal step fro ( 1, j) to(, j), the last colu will have height j ad ca be tiled (without startig with a square) i j 1 ways. The rest of the tilig cosists of a tiled lattice path fro (0, 0) to ( 1, j) which ca be created i 1+ j 1 ways. (Note that whe j 1, the suad is 0, sice 0 0, as is appropriate sice the last colu ca t have height 1 without startig with a square; also, whe j, 1 1, so the suad siplifies to +1, as required.) All together, the uber of tiligs is j0 j j j 1 1, as desired. By the exact sae logic, usig bracelet tiligs, we get Idetity 9. or, 1, j0 1+ j L j L j j 1 Replacig with ad replacig L with V, the last two idetities are appropriately colorized as well.

7 ARTHR T. BENJAMIN AND ELIZABETH REILAND 4. Ope Probles What follows is a list of ibooial idetities that are still i eed of cobiatorial proof. Soe of these idetities have extreely siple algebraic proofs (ad soe hold for ore geeral sequeces tha ibooial sequeces) so oe would expect the to have eleetary cobiatorial proofs as well. May siple idetities appear i iboacci Quarterly articles by Gould [4, 5]. j j j j j j j 1 1 j Here is aother basic idetity for geeralized bioial coefficiets, first oted by oteé [] ad further developed by Trojovsý [10] Here are soe alteratig su idetities, provided by Lid [7] ad Cooper ad Keedy [2], respectively, that ight be aeable to sig-reversig ivolutios: ( 1) j(j+1)/2 j j0 j0 1 ( 1) j(j+1)/2 j 1 j 0. Here are soe special cases of very itriguig forulas that appear i a recet paper by Kilic, Aus ad Ohtsua [] ( 1) 2 0 L 2 ( 1) 2 1 L We have just scratched the surface here. There are coutless others! Refereces [1] A. T. Bejai ad J. J. Qui. Proofs That Really Cout: The Art of Cobiatorial Proof, The Dolciai Matheatical Expositios, 27, Matheatical Associatio of Aerica, Washigto DC, 200. [2] C. Cooper ad R. E. Keedy. Proof of a Result by Jarde by Geeralizig a Proof by Carlitz, ib. Quart.4 (1995) [] G. oteé. Gééralisatio d ue orule Coue, Nouvelle.A.Math.,15 (1915), 112. [4] H. W. Gould. The bracet fuctio ad the oteé-ward Geeralized Bioial Coefficiets with Applicatio to ibooial Coefficiets, ib. Quart. 7 (199) 2 40, 55. [5] H. W. Gould. Geeralizatio of Herite s Divisibility Theores ad the Ma-Shas Priality Criterio for s-ibooial Arrays, ib. Quart. 12 (1974)

8 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES 7 [] E. Kilic, A. Aus, ad H. Ohtsua. Soe Geeralized ibooial Sus related with the Gaussia q- Bioial sus, Bull.Math.Soc.Sci.Math.Rouaie,55 (2012), [7] D. A Lid. A Deteriat Ivolvig Geeralized Bioial Coefficiets, ib. Quart 9.2 (1971) , 12. [8] E. Reilad. Cobiatorial Iterpretatios of ibooial Idetities, Seior Thesis, Harvey Mudd College, Clareot, CA [9] B. Saga ad C. Savage. Cobiatorial Iterpretatios of Bioial Coefficiet Aalogues Related to Lucas Sequeces, Itegers 10 (2010), [10] P. Trojovsý. O soe Idetities for the ibooial Coefficiets via Geeratig uctio, Discrete Appl. Math.155 (2007) o. 15, AMS Classificatio Nubers: 05A19, 11B9 Departet of Matheatics, Harvey Mudd College, Clareot, CA E-ail address: bejai@hc.edu Departet of Applied Matheatics ad Statistics, Johs Hopis iversity, Baltiore, MD E-ail address: ereilad@jhu.edu

2.6 Rational Functions and Their Graphs

2.6 Rational Functions and Their Graphs .6 Ratioal Fuctios ad Their Graphs Sectio.6 Notes Page Ratioal Fuctio: a fuctio with a variable i the deoiator. To fid the y-itercept for a ratioal fuctio, put i a zero for. To fid the -itercept for a