COMBINATORIAL PROOFS OF FIBONOMIAL IDENTITIES

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1 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES ARTHR T. BENJAMIN AND ELIZABETH REILAND Abstrct. ibooil coefficiets re defied lie bioil coefficiets, with itegers replced by their respective ibocci ubers. or exple, Rerbly, is l iteger. I 2010, Bruce Sg d Crl Svge derived two very ice cobitoril iterprettios of ibooil coefficiets i ters of tiligs creted by lttice pths. We believe tht these iterprettios should led to cobitoril proofs of ibooil idetities. We provide list of siple looig idetities tht re still i eed of cobitoril proof. 1. Itroductio Wht do you get whe you cross ibocci ubers with bioil coefficiets? ibooil coefficiets, of course! ibooil coefficiets re defied lie bioil coefficiets, with itegers replced by their respective ibocci ubers. Specificlly, for 1, or exple, ,5. ibooil coefficiets reseble bioil coefficiets i y. Alogous to the Pscl Trigle boudry coditios 1 d 1, we hve 1 d 1. We lso defie 0 1. Sice +( ) , Pscl s recurrece the followig log hs Idetity 1. or 2, As iedite corollry, it follows tht for ll 1, is iteger. Iterestig iteger qutities usully hve cobitoril iterprettios. or exple, the bioil coefficiet couts lttice pths fro (0, 0) to (, b) (sice such pth tes + b steps, of which re horizotl steps d the reiig b steps re verticl). As described i [1] d elsewhere, the ibocci uber +1 couts the to tile strip of legth with squres (of legth 1) d doios (of legth 2). As we ll soo discuss, ibooil coefficiets cout, ppropritely eough, tiligs of lttice pths! 2. Cobitoril Iterprettios I 2010 [9], Bruce Sg d Crl Svge provided two elegt coutig probles tht re euerted by ibooil coefficiets. The first proble couts restricted lier tiligs d the secod proble couts urestricted brcelet tiligs s described i the ext two theores. 1

2 2 ARTHR T. BENJAMIN AND ELIZABETH REILAND Theore 2. or, b 1, couts the to drw lttice pth fro (0, 0) to (, b), the tile ech row bove the lttice pth with squres d doios, the tile ech colu below the lttice pth with squres d doios, with the restrictio tht the colu tiligs re ot llowed to strt with squre. Let s use the bove theore to see wht is coutig. There re 20 lttice pths fro (0, 0) to (, ) d ech lttice pth cretes iteger prtitio ( 1, 2, )where 1 2 0, where i is the legth of row i. Below the pth the colus for copleetry prtitio ( 1, 2, )where or exple, the lttice pth below hs horizotl prtitio (, 1, 1) d verticl prtitio (0, 2, 2). The first row c be tiled 4 (ely sss or sd or ds where s deotes squre d d deotes doio). The ext rows ech hve oe tilig. The colus, of legth 0, 2 d 2 c oly be tiled i 1 wy with the epty tilig, followed by tiligs d d d sice the verticl tiligs re ot llowed to begi with squre. or other exple, the lttice pth ssocited with prtitio (, 2, 2) (with copleetry verticl prtitio (0, 0, 2)) c be tiled 12. These lttice pths re show below. (,) (,) 1wy 1wy 1 1 w w y y w y (0,0) (0,0) igure 1. The rows of the lttice pth (, 1, 1) c be tiled. The colus below the lttice pth, with verticl prtitio (0, 2, 2) c be tiled 1 wy sice those tiligs y ot strt with squres. This lttice pth cotributes tiligs to. The lttice pth (, 2, 2) cotributes 12 tiligs to The lttice pth ssocited with (, 1, 0) hs o legl tiligs sice its verticl prtitio is (1, 2, 2) d there re o legl tiligs of the first colu sice it hs legth 1. There re 10 lttice pths tht yield t lest oe vlid tilig. Specificlly, the pths ssocited with horizotl prtitios (,, ), (, 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 geerlly, for the ibooil coefficiet,wesuoverthe lttice pths fro (0, 0) to (, b) which correspods to iteger prtitio ( 1, 2,..., b )where 1 2 b 0, d hs correspodig verticl prtitio ( 1, 2,..., )where b. Recllig tht 0 0 d 1 1, this lttice pth cotributes b tiligs to. The secod cobitoril iterprettio of ibooil coefficiets utilizes circulr tiligs, or brcelets. A brcelet tilig is just lie lier tilig usig squres d doios, but brcelets.

3 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES lso llow doio to cover the first d lst cell of the tilig. As show i [1], for 1, the Lucs uber L couts brcelet tiligs of legth. or exple, there re L 4 tiligs of legth, ely sss, sd, ds d d s where d deotes doio tht covers the first d lst cell. Note tht L 2 couts ss, d d d where the d tilig is sigle doio tht strts t cell 2 d eds o cell 1. or cobitoril coveiece, we sy there re L 0 2epty tiligs. The ext cobitoril iterprettio of Sg d Svge hs the dvtge tht there is o restrictio o the verticl tiligs. Theore. or, b 1, 2 couts the to drw lttice pth fro (0, 0) to (, b), the ssig brcelet to ech row bove the lttice pth d to ech colu below the lttice pth. Specificlly, the lttice pth fro (0, 0) to (, b) tht geertes the prtitio ( 1, 2,..., b ) bove the pth d the prtitio ( 1, 2,..., ) below the pth cotributes L 1 L 2 L b L 1 L 2 L brcelet tiligs to 2. Note tht ech epty brcelet cotributes fctor of 2 to this product. or exple, the lttice pth fro (0, 0) to (, ) with prtitio (, 1, 1) bove the pth d (0, 2, 2) below the pth cotributes L L 1 L 1 L 0 L 2 L 2 72 brcelet tiligs euerted by (,) L 4 1wy 1wy (0,0) L 0 2 igure 2. The rows bove the lttice pth c be tiled with brcelets i 4 d the colus below the pth c be tiled with brcelets i L 0 L 2 L This cotributes 72 brcelet tiligs to I their pper, Sg d Svge exted their iterprettio to hdle Lucs sequeces, defied by 0 0, 1 1 d for 2, 1 + b 2. Here +1 euertes the totl weight of ll tiligs of legth where the weight of tilig with i squres d j doios is i b j. (Altertively, if d b re positive itegers, +1 couts colored tiligs of legth where there re colors for squres d b colors for doios.) Liewise the uber of weighted brcelets of legth is give by V V 1 + bv 2 with iitil coditios V 0 2 d V 1 (so the epty brcelet hs weight of 2). This leds to cobitoril iterprettio of Lucsoil coefficiets, defied lie the ibooil coefficiets. or exple, Both of the previous cobitoril iterprettios wor exctly s before, usig weighted (or colored) tiligs of lttice pths.

4 4 ARTHR T. BENJAMIN AND ELIZABETH REILAND. Cobitoril Proofs Now tht we ow wht they re coutig, we should be ble to provide cobitoril proofs of ibooil coefficiet idetities. or exple, Idetity 1 c be rewritte s follows. Idetity 4. or, 1, Cobitoril Proof: The left side couts tiligs of lttice pths fro (0, 0) to (, ). How y of these tiled lttice pths ed with verticl step? As show below, i ll of these lttice pths, the first row hs legth d c be tiled +1. The rest depeds o the lttice pth fro (0, 0) to (, 1). Suig over ll possible lttice pths fro (0, 0) to (, 1) there re + 1 tiled lttice pths for the rest of the lttice. Hece the uber of tiled lttice pths edig i verticl step is (, ) (, 1) (0, 0) igure. There re tiled lttice pths tht ed with verticl step. How y tiled lttice pths ed with horizotl step? I ll such pths, the lst colu hs legth d c be tiled 1 (begiig with doio). Suig over ll lttice pths fro (0, 0) to ( 1,) there re tiled lttice pths for the rest of the lttice. Hece the uber of tiled lttice pths edig i horizotl step, s illustrted below, is (0, 0) tiled lttice pths tht ed with hori- igure 4. There re zotl step. ( 1,) (, ) 1 doio Cobiig the two previous cses, the totl uber of tiled lttice pths fro (0, 0) to (, ) is

5 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES 5 Replcig lier tiligs with brcelets d reovig the iitil doio restrictio for verticl tiligs, we c pply the se logic s before to get L L. 1 Dividig both sides by gives us Idetity 5. or, 1, L + L. 1 I full disclosure, Idetities 4 d 5 re used by Sg d Svge to prove their cobitoril iterprettios, so it is o surprise tht these idetities would hve esy cobitoril proofs. The se is true for the weighted (or colorized) versio of these idetities for Lucsoil coefficiets. Idetity. or, 1, + Idetity 7. or, 1, V + V. 1 By cosiderig the uber of verticl steps tht lttice pth eds with, Reild [8] proved Idetity 8. or, 1, + j 1+ j +1 j 1 1 j0 Cobitoril Proof: We cout the tiled lttice pths fro (0, 0) to (, ) by cosiderig the uber j of verticl steps t the ed of the pth, where 0 j. Such tilig begis with j full rows, which c be tiled j +1. Sice the lttice pth ust hve horizotl step fro ( 1, j) to(, j), the lst colu will hve height j d c be tiled (without strtig with squre) i j 1. The rest of the tilig cosists of tiled lttice pth fro (0, 0) to ( 1, j) which c be creted i 1+ j 1. (Note tht whe j 1, the sud is 0, sice 0 0, s is pproprite sice the lst colu c t hve height 1 without strtig with squre; lso, whe j, 1 1, so the sud siplifies to +1, s required.) All together, the uber of tiligs is j0 j j j 1 1, s desired. By the exct se logic, usig brcelet tiligs, we get Idetity 9. or, 1, j0 1+ j L j L j j 1 Replcig with d replcig L with V, the lst two idetities re ppropritely colorized s well.

6 ARTHR T. BENJAMIN AND ELIZABETH REILAND 4. Ope Probles Wht follows is list of ibooil idetities tht re still i eed of cobitoril proof. Soe of these idetities hve extreely siple lgebric proofs (d soe hold for ore geerl sequeces th ibooil sequeces) so oe would expect the to hve eleetry cobitoril proofs s well. My siple idetities pper i ibocci Qurterly rticles by Gould [4, 5]. j j j j j j j 1 1 j Here is other bsic idetity for geerlized bioil coefficiets, first oted by oteé [] d further developed by Trojovsý [10] Here re soe ltertig su idetities, provided by Lid [7] d Cooper d Keedy [2], respectively, tht ight be eble to sig-reversig ivolutios: ( 1) j(j+1)/2 j j0 j0 1 ( 1) j(j+1)/2 j 1 j 0. Here re soe specil cses of very itriguig foruls tht pper i recet pper by Kilic, Aus d Ohtsu [] ( 1) 2 0 L 2 ( 1) 2 1 L We hve just scrtched the surfce here. There re coutless others! Refereces [1] A. T. Beji d J. J. Qui. Proofs Tht Relly Cout: The Art of Cobitoril Proof, The Dolcii Mtheticl Expositios, 27, Mtheticl Associtio of Aeric, Wshigto DC, 200. [2] C. Cooper d R. E. Keedy. Proof of Result by Jrde by Geerlizig Proof by Crlitz, ib. Qurt.4 (1995) [] G. oteé. Géérlistio d ue orule Coue, Nouvelle.A.Mth.,15 (1915), 112. [4] H. W. Gould. The brcet fuctio d the oteé-wrd Geerlized Bioil Coefficiets with Applictio to ibooil Coefficiets, ib. Qurt. 7 (199) 2 40, 55. [5] H. W. Gould. Geerliztio of Herite s Divisibility Theores d the M-Shs Prility Criterio for s-ibooil Arrys, ib. Qurt. 12 (1974)

7 COMBINATORIAL PROOS O IBONOMIAL IDENTITIES 7 [] E. Kilic, A. Aus, d H. Ohtsu. Soe Geerlized ibooil Sus relted with the Gussi q- Bioil sus, Bull.Mth.Soc.Sci.Mth.Rouie,55 (2012), [7] D. A Lid. A Deterit Ivolvig Geerlized Bioil Coefficiets, ib. Qurt 9.2 (1971) , 12. [8] E. Reild. Cobitoril Iterprettios of ibooil Idetities, Seior Thesis, Hrvey Mudd College, Clreot, CA [9] B. Sg d C. Svge. Cobitoril Iterprettios of Bioil Coefficiet Alogues Relted to Lucs Sequeces, Itegers 10 (2010), [10] P. Trojovsý. O soe Idetities for the ibooil Coefficiets vi Geertig uctio, Discrete Appl. Mth.155 (2007) o. 15, AMS Clssifictio Nubers: 05A19, 11B9 Deprtet of Mthetics, Hrvey Mudd College, Clreot, CA E-il ddress: beji@hc.edu Deprtet of Applied Mthetics d Sttistics, Johs Hopis iversity, Bltiore, MD E-il ddress: ereild@jhu.edu