Two-Toed Tiligs ad Compositios of Iteges Melaie Hoffma Abstact. Followig the aticle Combiatoics of Two-Toed Tiligs by Bejami, Chi, Scott, ad Simay [1], this pape itoduces a fuctio to cout tiligs of legth + that use ay umbe of white tiles (of legth betwee 1 ad ) ad exactly idetical ed squaes. We exploe a umbe of combiatoial idetities ad geealizatios of this cocept, alog with some coectios to geealized Fiboacci umbes ad applicatios to compositios of iteges. A Math 501 Poject ude the diectio of D. Joh S. Caughma Submitted i patial fulfillmet of the equiemets fo the degee of Maste of Sciece i Mathematics at Potlad State Uivesity Decembe 4, 2017
Cotets I. Itoductio II. Two-toed Tiligs III. Combiatoial Idetities IV. Geealizatios V. Applicatios to Compositios VI. Coclusio VII. Refeeces
I. Itoductio Oe of the simple topics studied i combiatoics is the sequece of Fiboacci umbes, itoduced by Leoado de Pisa, a ifluetial Italia mathematicia fom the Middle Ages who is commoly kow by the ame Fiboacci. This sequece of umbes appeas fequetly i atue - fo example, i the family tees of hoeybees, the umbe of petals i some flowes, the spial pattes of seed heads ad pie coes, the aagemet of leaves o stems, ad so o. Fiboacci umbes also show up i the mete of Saskit poety ad i at, music, ad achitectue. The Fiboacci umbes ae defied ecusively by settig F 1 = F 2 = 1 ad lettig F = F - 1 + F - 2 fo 3. Oe easily poved fact, which we will late establish, is that the Fiboacci umbe F m + 1 couts the umbe of tiligs of a m by 1 ectagle (heeafte called a m - stip) usig oly squaes ad domioes. At the othe exteme, oe of the moe subtle topics studied i combiatoics is how to cout the umbe of patitios of a itege (i.e., the umbe of ways of witig some positive itege as a sum of positive iteges). Fo istace, 7 is the umbe of itege patitios of 5, sice 5 may be witte as ay of the followig seve expessios 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, ad 1 + 1 + 1 + 1 + 1. Note that the ode of the summads is disegaded, so 3 + 2 is cosideed equivalet to 2 + 3 as a itege patitio of 5. It is vey difficult to obtai a closed-fom fomula fo the umbe of itege patitios. Famously, mathematicias Hady ad Ramauja poved i 1918 that the 1 umbe of patitios of appoaches exp ( π ) as. 4 3 2 3 Vaious questios coceig itege patitios have bee exploed ove the cetuies. Fo example, if we ageed to distiguish sums by the ode of thei tems, the 5 + 2 ad 2 + 5 would be cosideed diffeet fom each othe, ad these two odeed patitios of 7 would be couted sepaately. Such a odeed patitio is called a compositio of. We may wat to cout the compositios of with exactly p pats (i.e., summads) of size k. Fo istace, thee ae fou compositios of 7 with exactly thee pats of size 2: 2 + 2 + 2 + 1, 2 + 2 + 1 + 2, 2 + 1 + 2 + 2, ad 1 + 2 + 2 + 2. This poject exploes these ideas futhe, examiig the aticle Combiatoics of two-toed tiligs by Bejami, Chi, Scott, ad Simay [1]. Thei pape itoduces seveal iteestig idetities ivolvig tiligs, icludig elatioships with geealized Fiboacci umbes ad compositios of iteges.
II. Two-toed Tiligs Fo oegative iteges ad, let the fuctio a (, ) cout the umbe of two-toed tiligs of a stip of legth + cosistig of exactly ed squaes ad ay umbe of white tiles of ay legth (fom 1 to ). To abbeviate, we will call these (, ) - tiligs. The possible (1, 2) - tiligs ae R11, 1R1, 11R, R2, 2R, whee R sigifies a ed squae ad a umbe sigifies a white tile of that legth. Visually, these tiligs may be epeseted as follows. This eumeatio demostates that a (1, 2) = 5. O the othe had, a (2, 1) = 3 sice RR1, R1R, ad 1RR ae all the possible (2, 1) - tiligs. I geeal, a (, 1) = + 1 sice thee ae + 1 potetial positios fo the white tile (i this case a squae) ad ed squaes must fill the othe positios. We begi with iitial coditios ad a ecuece elatio fo a (, ). Idetity 1: Fo 0, the umbe of (, ) - tiligs satisfies a (, 0) = 1. Fo 1, a (0, ) = 2-1. Fo, 1, a (, ) = a ( 1, ) + 2 a (, 1) a ( 1, 1).
Poof : If thee ae o white tiles, all tiles must be ed squaes, so a (, 0) = 1. If thee ae o ed squaes, we may tile a - stip with white tiles by decidig whethe o ot to ed a tile at evey cell except the fial oe (which must ed a tile). Thus a (0, ) = 2-1, ad we ote that this expessio also coespods vey atually to the umbe of compositios (i.e., odeed patitios) of the positive itege. To cout (, ) - tiligs fo positive values of ad, we coditio o the way the (, ) - tilig eds. If it eds with a ed squae, thee ae a ( 1, ) ways to tile the pevious 1 + cells. If it eds with a white squae, thee ae a (, 1) ways to tile the pevious + 1 cells. If it eds with a white tile of legth geate tha 1, we may obtai it fom a (, 1) - tilig that eds i a white tile by legtheig the last tile by 1. Thee ae a (, 1) a ( 1, 1) of these (all (, 1) - tiligs except those edig i a ed squae). Sice these ae the oly possibilities, we have a (, ) = a ( 1, ) + 2 a (, 1) a ( 1, 1). Usig Idetity 1, we ca fill i a table of a (, ) fo values of ad betwee 0 ad 5, iclusive. Whe = 0, we have a (, 0) = 1, which gives us evey ety of the fist colum of the table. Whe = 0 ad 1, we have a (0, ) = 2-1, which gives us the est of the fist ow of the table. We also kow a (, 1) = + 1, which gives us the est of the secod colum. Fo each emaiig ety, we use a (, ) = a ( 1, ) + 2 a (, 1) a ( 1, 1), which coespods to that ety s Noth eighbo, plus twice its West eighbo, mius its Nothwest eighbo. Fom above, we kow a (1, 2) = 5 ad a (2, 1) = 3, so we use the ecusio to calculate a (2, 2): a (1, 2) + 2 a (2, 1) a (1, 1) = 5 + 2 2 = 9. The a (3, 2) = 9 + 2(4) 3 = 14, ad so o. 0 1 2 3 4 5 0 1 1 2 4 8 16 1 1 2 5 12 28 64 2 1 3 9 25 66 168 3 1 4 14 44 129 360 4 1 5 20 70 225 681 5 1 6 27 104 363 1182 Table 1: Two-toed tiligs a (, )
III. Combiatoial Idetities The Fiboacci umbes 1, 1, 2, 3, 5, 8, appea as sums of the diagoals of Pascal s tiagle, idicated i colo below. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 I a simila mae, ou ext idetity elates sums of the diagoals of the table fo two-toed tiligs (Table 1) to the odd-idexed Fiboacci umbes. As a example, a(0, 2 ) + a(1, 1 ) + a(2, 0 ) = 2 + 2 + 1 = 5 = F 5. Idetity 2: Fo 0, = 0 a(, ) = F 2 + 1. Poof : The left side eumeates the two-toed tiligs of legth with ed squaes whee may vay fom 0 to. We wish to defie a map g betwee the set of two-toed tiligs of legth ad the set of tiligs of a stip of legth 2 usig white squaes ad domioes. To do so, we fist defie a map g which is the estictio of g to the idividual tiles belogig to a two-toed tilig of legth. Specifically, let the fuctio g map each ed squae to a white domio (i.e., R 2) ad each white tile of legth k 1 to k 1 white domioes, peceded ad followed by a white squae (i.e., k 122...21 whee thee ae k 1 copies of 2). I paticula, g maps each white squae to two white squaes (i.e., 1 11). To costuct g, apply g to each tile of a two-toed tilig fom left to ight util eachig the ed, cocateatig the esults. Fo ay two-toed tilig y 1 y 2...y h of legth whee the y i epeset idividual tiles, we have g (y 1 y 2...y h ) = g (y 1 ) g (y 2 )... g (y h ). The fuctio
g effectively takes ay two-toed tilig, doubles its legth ad chages it all to white, usig oly squaes ad domioes. We will show that g is a bijectio. To show g is oto, we will begi with a white tilig of a 2 - stip usig white squaes ad domioes ad idetify the two-toed tilig which is its peimage ude g. Give a abitay white tilig, its leftmost block of tiles must cosist of oe of the followig ad have the give peimage: some umbe d of domios peceded by zeo white squaes. The peimage is d ed squaes (i.e., RR...R 22...2 whee the umbe of R s ad 2 s ae the same). a eve umbe 2 t of cosecutive white squaes fo some t 1. The peimage is t cosecutive white squaes. a odd umbe 2 t + 1 of cosecutive white squaes, t 0, followed by d 1 white domioes ad aothe white squae. (Note that the tilig of a 2 - stip caot cosist solely of a odd umbe white squaes sice 2 is eve.) The peimage is t cosecutive white squaes, the a white tile of legth d + 1. Fo example, if we have oe white squae followed by d 1 white domioes ad aothe white squae, the peimage is a white tile of legth d + 1 (i.e., d + 1 122...21 with d copies of 2). Afte fidig the peimage of the leftmost block of tiles, move o to the ext leftmost block of tiles ad fid its peimage. Each such block will be of eve legth ad its peimage will be a two-toed tilig of half its legth ad will be uiquely detemied by the ules above. Sice we may decostuct evey possible tilig of a 2 - stip ito such blocks with the specified peimages, we have show g is oto. Next, we pove g is oe-to-oe. Suppose two distict two-toed tiligs Y ad Z of legth ae mapped by g to the same tilig of the 2 - stip; i.e., g (Y) = g (Z). Let Y = y 1 y 2...y h ad Z = z 1 z 2...z j whee y 1,, y h, z 1,, z j ε {1, 2,,, R}. By way of cotadictio, assume Y Z. The thee exists some fist tile k i which Y ad Z diffe, so that y k z k ad y m = z m fo all m such that 1 m < k. But the g (y k ) g (z k ), which implies g (Y) g (Z), a cotadictio. So g is oe-to-oe ad thus a bijectio. Fially, we wat to show that F 2 + 1 couts the set of all tiligs of a 2 - stip usig white squaes ad domioes, which will complete the poof of the idetity. Let t m epeset the umbe of tiligs of a m - stip usig white tiles of legth 1 o 2. To cout these, we coditio o the ightmost tile. If it is a white squae, thee ae t m - 1 ways to tile the emaiig m 1 cells. If it is a white domio, thee ae t m - 2 ways to tile the emaiig m 2 cells. Thus fo m 2, we have poved the ecuece elatio t m = t m - 1 + t m - 2. This is idetical to the Fiboacci ecuece F m = F m - 1 + F m - 2. Next, compae the iitial coditios. We kow F 1 = F 2 = 1. A stip with o legth has the empty tilig ad a 1-stip may be tiled oly with a white squae, so
t 0 = t 1 = 1. A 2-stip may be tiled usig two squaes o oe domio, so t 2 = 2. So t m = F m + 1 fo ay m 0, which shows that the umbe t 2 of tiligs of a 2 -stip, is F 2 + 1. The ext thee idetities ivolve biomial coefficiets ad aise fom costuctig ou two-toed tiligs i vaious maes. Fist, suppose we wat to cout two-toed tiligs by coditioig o the umbe of white tiles. Idetity 3: Fo 0 ad 1, a (, ) = ( ) ( ). j = 1 1 j + Poof : The left side couts the umbe of (, ) - tiligs. We may also cout these tiligs by coditioig o j, the umbe of white tiles, whee 1 j. Fo each value of j, we fist detemie the legths of the white tiles that compise legth. Sice the j th tile must ed at the th 1 cell, thee ae ( ) ways to choose othe cells of the emaiig 1 cells whee a white tile eds. We ow have white tiles w 1, w 2,, w j. Thus ( exactly j summads. 1 ) couts compositios of with Next, we itespese ed squaes amog the white tiles i thei give ode. Altogethe, j + thee ae j + positios fo tiles, so ( ) ways to choose whee to put the ed squaes. By the 1 j + poduct popety, we have ( ) ( ) two-toed tiligs of legth + with exactly j white tiles. Alteatively, i the secod step, we may coside the j + 1 gaps befoe ad afte the j j + 1 white tiles as locatios to which we may assig ed squaes i (( )) ways. The j + 1 givig the same poduct as befoe. j + 1 + 1 j + (( )) = ( ) = ( ), Now suppose we wat to cout two-toed tiligs by coditioig o the umbe of locatios amog the ed squaes whee white tiles occu. Idetity 4: Fo 0 ad 1, + 1 j = 1 + 1 1 j a (, ) = ( ) ( ) 2 - j. Poof : Agai, the left side eumeates the umbe of (, ) - tiligs. Aothe way to cout these is to begi with ed squaes ad choose fom the + 1 gaps befoe ad afte these ed squaes
exactly j egios i which to place white squaes. (Sice 1, thee must be at least 1 such + 1 ( j egio ad at most + 1 egios, so 1 j + 1.) We may choose j egios i ) ways. Next, select a subset { x 1, x 2,, x j - 1 } of {1, 2,, 1} such that 1 x 1 < x 2 < < x j - 1 <. 1 Thee ae ( ) such subsets. Each elemet x i of a paticula subset gives the patial sum of white squaes placed i the fist i egios; i.e., put x 1 white squaes i egio 1, x i x i - 1 white squaes i egio i fo 2 i j - 1, ad x j - 1 white squaes i egio j. Fially, decide which white squaes to attach o lik togethe with the squae o thei left. Thee ae white squaes, j of which ae leftmost i thei egio, so 2 - j ways to decide whethe o ot to attach the - j emaiig - j white squaes to thei left eighbo. By the poduct piciple, ( + 1 j ) ( 1 ) 2 is the umbe of (, ) - tiligs with exactly j egios whee white tiles occu, which we sum ove all possible values of j. The fial idetity coceig two-toed tiligs gives a simila expessio, but with a facto of 2 befoe the summatio. Idetity 5: Fo 0 ad 1, + 1 ( j j = 0 j + a (, ) = 2 - - 1 ) ( ). Poof : We make use of thee diffeet types of two-toed tiligs. Fist, let T be the total set of all (, ) - tiligs. The T = a (, ). Next, let S epeset the set of all (, ) - tiligs that cosist + solely of squaes. Sice the oly questio is which of the + squaes ae ed, S = ( ). Fially, let D epeset squae-oly (, ) - tiligs i which we may place ed divides o ay boudaies ot adjacet to a white squae (i.e., befoe cell 1 ad/o afte cell + if it is ed, ad/o betwee two cosecutive ed squaes). The D is a decoated vesio of S, ad we say the divides ae white-avese. Fo example, if = 8 ad = 4, oe elemet of D is the tilig RR w R R RR ww R w R with ed divides befoe the fist, fifth, ad sixth cells, ad afte the twelfth cell; aothe is R R w RRR R ww R w R with ed divides befoe the secod ad seveth cells, ad afte the twelfth cell. To cout tiligs i D, let s coditio o j, the umbe of ed divides. If j wee allowed to be + 1, oe of the gaps befoe ad afte the ed squaes could have ay white squaes, sice the divides ae white-avese. But 1, so thee ae at most ed divides, ad 0 j. We + 1 ( j j + claim the umbe of squae-oly (, ) - tiligs with exactly j ed divides is ) ( ). To see this, stat with ed squaes ad place ed divides i j of the + 1 gaps befoe ad afte + 1 them i ) ways. Sice o white squae may be ext to a divide, thee emai + 1 j ( j
+ 1 j + 1 j + 1 egios whee we may distibute white squaes i (( )) = ( ) = ( ways. So + 1 ( j j = 0 j + D = ) ( ). j + ) We still eed to show that a (, ) = 2 - - 1 D. Fo k 0, let S k be the set of tiligs i S with exactly k boudaies ot adjacet to ay white squae. Fo example, if = 8 ad = 4, the tilig RR w RRRR ww R w R has k = 6. Similaly, we may defie espective subsets T k ad D k of T ad D with k o-white boudaies. Sice D k diffes fom S k i that we may choose whethe o ot to place ed divides at ay of the o-white boudaies, D k = 2 k S k. Note that if = 0 o all ed squaes ae suouded o both sides by white squaes, we have k = 0. Just as j, the umbe of ed divides, is bouded by, so is k, the umbe of o-white boudaies whee we may put ed divides, bouded by. Next, let v = the umbe of boudaies betwee adjacet white squaes. The v is a fuctio of k. Give some fixed k, we may tu a tilig i S k ito a tilig i T k by decidig at each white/white bouday whethe o ot to joi the 2 squaes togethe to make a loge tile. Fo istace, the tilig RR w RRRR ww R w R i S 6 has v = 1 so ceates 2 tiligs i T 6 : oe with all squaes as show ad the othe with a white domio i place of ww. So T k = 2 v S k. I S k, thee ae + tiles ad a total of + + 1 boudaies. We have fou types of boudaies as we ead a tilig i S k fom left to ight: ed/ed, ed/white, white/ed, ad white/white. The walls befoe cell 1 ad afte cell + ae coloed ed. Afte the left wall ad afte each of the ed squaes, thee is eithe a ed/ed o a ed/white bouday. Sice k of these ae ed/ed, that leaves + 1 k ed/white boudaies. Similaly, befoe each of the ed squaes ad befoe the ight wall, thee is eithe a ed/ed o a white/ed bouday. Sice k of these ae ed/ed, that leaves + 1 k white/ed boudaies. Summig all fou types of boudaies, we have k + 2( + 1 k ) + v = + + 1. Solvig fo v, we get v = 1 + k. By substitutio, T k = 2 - - 1 + k S k = 2 - - 1 + k ( D k / 2 k ) = 2 - - 1 D k. We wat T, the total umbe of two-toed (, ) - tiligs. This is just the sum ove all oegative k values of two-toed (, ) - tiligs with k ed / ed boudaies: a (, ) = T = T k = 2 - - 1 D k = 2 - - 1 D k = 2 - - 1 D. k 0 k 0 k 0 IV. Geealizatios Next, we estict ouselves to two-toed tiligs that ed with at least s white tiles. These so-called (,, s ) - tiligs of legth + + s have ed squaes ad white tiles of total legth
+ s. So s has a dual pupose, cotibutig both to the total legth of white tiles ad also specifyig the miimum umbe of white tiles with which the tilig eds. Fo example, coside the possible (1, 2, 1) - tiligs: R111, R12, R21, R3, 1R11, 1R2, 2R1, ad 11R1. So a (1, 2, 1) = 8. Note that whe s = 0, the umbe of (,, s ) - tiligs is equivalet to the umbe of (, ) - tiligs; i.e., a (,, 0) = a (, ). Idetity 6: Fo, 0 ad s 1, a (,, s ) = a (, j, s 1 ). j = 0 Poof : The left side is the umbe of (,, s ) - tiligs. Let j be the legth of the last tile (white of positive legth sice s 1). Now + s is the total legth of white tiles. Sice ou tilig eds with at least s white tiles, thee ae at least s 1 cells used i the othe edig white tiles, so j has maximum legth + s ( s 1) = + 1. If we emove the fial tile of legth j, the pecedig + + 1 j = 1 + s j cells may be tiled i a (, + 1 j, s 1 ) ways. So a (, + 1 j, s 1 ) = a (,, s ). Now whe j = 1, + 1 j =, ad whe j = + 1, + 1 j = 0, so we may eplace the secod agumet + 1 j with j ad idex fom 0 to : a (,, s ) = a (, j, s 1 ). j = 0 Fo example, to calculate a (1, 2, 2), we eed to fid some iitial values. Note that the oly (1, 0, 1) - tilig is R1, so a (1, 0, 1) = 1; R11, R2 ad 1R1 ae the oly (1, 1, 1) - tiligs, so a (1, 1, 1) = 3. Usig Idetity 6 ad ou pevious wok, we get a (1, 2, 2) = a (1, 0, 1) + a (1, 1, 1) + a (1, 2, 1) = 1 + 3 + 8 = 12. We veify this esult by listig all (1, 2, 2) - tiligs: R1111, R112, R121, R211, R22, R13, R31, 1R111, 1R12, 1R21, 2R11, ad 11R11. Idetity 7: Fo 1 ad, s 0, a (,, s ) = a( 1, j, s + j ) j = 0 Poof : The left side is the umbe of (,, s ) - tiligs. Fo 0 j, suppose s + j gives the exact umbe of white tiles with which ay two-toed tilig of legth + + s eds. The those s + j white tiles ae immediately peceded by a ed squae which, if emoved, leaves a tilig of
legth ( 1 ) + + s that eds with at least s + j white tiles. We may cout the umbe of such tiligs by a( 1, j, s + j ). Summig ove all possible values of j yields the idetity. Idetity 8: Fo, s 0 ad 1, a (,, s ) = ( ( ). j = 0 + s 1 j + s 1 ) + j Poof : The left side is the umbe of (,, s ) - tiligs. We may costuct such a tilig by statig with exactly j + s white tiles, whee j may vay fom 0 (sice all white tiles may be at the ed) to (sice + s is the total legth of white tiles, ad each could be a squae). The total legth of white tiles is + s cells, ad the last cell must ed a tile, so we may choose the j + s 1 othe cells whee white tiles ed i ( + s 1 j + s 1 ) ways. Now we place the ed squaes. Because the tilig must ed with at least s white tiles, thee ae j + 1 locatios i which we may put the ed j + 1 + j + 1 1 + j squaes, so (( )) = ( ) = ( ) ways to do this. Applyig the poduct piciple ad summig ove all values of j yields the idetity. Coside the case i which ad s ae equal. Idetity 9: Fo 0 ad 1, + + 2 + a (,, ) = ( ) 2-1. Poof : Fist, we fid a equivalet expessio fo pat of the ight side usig the distibutive + + + popety ad the defiitio of biomial coefficiets: ( ) = ( )1 + ( ) + ( + )!! ( + )! + ( + 1)! + + ( 1)!! + + 1 1 + 2 + + 1 1 = ( ) + = ( ) + = ( ) + ( ). So ow we will pove the equivalet equatio a (,, ) = [( ) + ( )] 2-1. The left side couts two-toed tiligs of legth + 2 that have exactly ed squaes ad ed with at least white tiles. We tasfom this type of tilig ito aothe by eplacig each of the ed squaes oe by oe, fom left to ight, with the fial white tiles, ad the coloig these pik. Sice all ed squaes ae eplaced, this ew tilig is of legth + ad cosists of white ad pik tiles, exactly of which ae pik. Because we may udo these steps to get back the oigial two-toed tilig, these white ad pik tiligs ae also couted by a (,, ). + Next, we show aothe way to cout the white ad pik tiligs just descibed. Let X, a - subset of {1, 2,, + }, ame the cells i which pik tiles begi. We may choose this + subset X i ( ) ways. Coside whethe the vey fist tile is pik. Case 1: If the fist of pik + 1 tiles begis i cell 1, the 1 belogs to X ad thee ae ( 1 ) ways to choose the emaiig elemets of X. We may decide whethe each of the emaiig cells begis a white tile i 2 + + 1 ways. Case 2: If the fist pik tile does ot begi i cell 1, thee ae ( ) ( ) ways to 1
choose X ad so desigate the cells i which pik tiles begi. Cell 1 must begi a white tile, so thee emai 1 othe cells which may o may ot begi a white tile i 2-1 ways. Summig + 1 these two cases gives us ( ) 2 + + 1 1 + [( ) ( 1 )] 2-1, fom which we facto out 2-1 : + 1 + + 1 [( ) 2 + ( ) ( )] 2-1 + + 1 = [( ) + ( )] 2-1. 1 1 We ow itoduce the k th ode Fiboacci umbes. Give iitial coditios 1 F = 0 fo 0 ad = 1, we defie the th tem of the k th ode Fiboacci sequece ecusively fo 2: F 1 F = F - 1 + F - 2 +... + F - k. To illustate, F 2 F 3 F 4 = + + = 1 + 0 + 0 = 1; F 1 F 0 F 1 F 0 = + + = 1 + 1 + 0 = 2; F 2 F 1 = + + = 2 + 1 + 1 = 4; F 3 F 2 F 1 = + + = 4 + 2 + 1 = 7; etc. F 5 F 4 F 3 F 2 Thus the 3 d ode Fiboacci sequece begis 1, 1, 2, 4, 7,. Note that the 2 d ode Fiboacci sequece, i which each tem is the sum of the pevious two, is simply the familia Fiboacci sequece 1, 1, 2, 3, 5,. The ext idetity elates k th ode Fiboacci umbes ad two-toed tiligs. Idetity 10: Fo, k 1, F + 1 k + 1 = (-1) a (, ( k + 1), ). = 0 Poof : Fix k 1 ad let t be the umbe of tiligs of a -stip usig oly white tiles of legth k o less. Coditio o the legth of the last tile. We see that = + + +. This t t 1 t 2 t k ecuece matches that of the k th ode Fiboacci umbes. Cosideig the iitial coditios, if k 1 thee is oe way to tile a stip of zeo legth ad oe way to tile a 1-stip, i.e., t 0 = 1 = ad = 1 =. Thus =. F 1 t 1 F 2 t F + 1 Now we itepet the ight side of the idetity. We begi with all two-toed tiligs of legth k + that have exactly ed squaes ad ed with at least white tiles. These ae couted by a (, ( k + 1), ). Give such a tilig, if we legthe each of its last tiles by k, the eplace its ed squaes with the elogated white tiles oe by oe fom left to ight, the tilig is loge by copies of k ad shote by, so its legth is ow k + + k =. This white tilig of legth is guaateed to have tiles loge tha k i the positios that oigially
had ed squaes. To cout white tiligs of legth with tiles of maximum legth k, we apply the iclusio/exclusio piciple. The geatest allowable umbe of too log tiles (legth k + 1 o moe) is sice this would give us a total legth of ( k + 1). So vaies fom 0 k + 1 k + 1 to. Whe = 0, (-1) 0 a (0, 0( k + 1), 0) = a (0,, 0) = a (0, ), which couts tiligs of a k + 1 - stip usig white tiles of ay legth up to. By Idetity 1, a (0, ) = 2-1. Fom this total umbe of white tiligs usig tiles of legths up to, we subtact the white tiligs whee = 1, meaig those with at least oe tile loge tha k. The we add back all white tiligs whee = 2, meaig those with at least two tiles loge tha k. We cotiue to alteately subtact ad add i this mae, givig the sum o the ight side of the idetity. Coollay: Fo, k 1, = (-1) ( ) 2 - k - - 1. F + 1 k + 1 = 0 k k + k Poof : Stat with Idetity 10 ad use Idetity 9 to tasfom the ight side: F + 1 k + 1 = (-1) a (, ( k + 1), ) = (-1) ( ) 2 = 0 k + 1 = 0 k + 1 (k + 1) + (k + 1) + 2 (k + 1) + = (-1) ( ) 2 - k - - 1. = 0 k k + k - (k + 1) - 1 V. Applicatios to Compositios A compositio of is a odeed list of positive iteges that sum to. Fo example, the compositios of 4 ae 1111, 112, 121, 211, 22, 13, 31, ad 4; thee ae eight of them. A compositio of coespods to a ucoloed tilig of a -stip with tiles of legth 1 to, whee each summad is epeseted as a tile of positive itege legth. Let L( k, ) be defied as the umbe of compositios of i which at least 1 copy of the summad k appeas. Similaly, defie L p ( k, ) as the umbe of compositios of i which at least p copies of the summad k appea. Fo example, fom the list above, we see that L(2, 4) = 4 ad L 2 (2, 4) = 1. Idetity 11: Fo, k 1, L( k, ) = (-1) j - 1 a ( j, jk ). j 1
Poof : Now a ( j, jk ) couts two-toed tiligs of legth j + jk with exactly j ed squaes. If we eplace each ed squae with a pik tile of legth k, the tilig shiks by j ad gows by jk so is of legth. This coespods to a compositio of with j o moe istaces of the summad k, j of which ae witte i pik. Fo example, a (1, k ) stads fo all compositios of whee k is the fist summad (ad possibly othes), o k is the secod summad (ad possible othes), ad so o. Howeve, this is a ovecout sice a compositio with moe tha oe summad k will be couted i each of the positios at which k occus. To coect this, we use the iclusio-exclusio piciple. Sice j deotes the miimum umbe of occueces of the summad k, j has a value of at least 1. Note that oce j > the jk < 0 ad a ( j, jk ) = 0, k so the sum is fiite. Idetity 12: Fo p,, k 1, L p ( k, ) = (-1) j - p ( ) a ( j, jk ). j p p 1 Poof : By modifyig the iclusio-exclusio piciple somewhat, as show i Poofs that eally cout [2], we may cout the umbe of ways a popety occus at least p times. Idexig ove values of j p, we multiply each usiged summad of Idetity 11 by (-1) j - p ( ) to get the p 1 moe geealized fomula i Idetity 12. Notice that whe p = 1, this educes to Idetity 11. Let E p ( k, ) be defied as the umbe of compositios of with exactly p copies of the summad k. Fo example, the compositios of 4 with exactly 2 copies of the summad 1 ae 112, 121, ad 211, so E 2 (1, 4) = 3. Idetity 13: Fo p,, k 1, E p ( k, ) = (-1) j - p ( ) a ( j, jk ). j p j p Poof : If we wat the compositios of i which the summad k appeas exactly p times, we may stat with the compositios of i which the summad k appeas at least p times ad subtact the compositios of i which the summad k appeas at least p + 1 times. That is, E p ( k, ) = L p ( k, ) L p + 1 ( k, ) = (-1) j - p ( ) a ( j, jk ) (-1) j - p - 1 ( ) a ( j, jk ) by Idetity 12 j p p 1 j p + 1 = (-1) j - p ( ) a ( j, jk ) + (-1) j - p ( ) a ( j, jk ) by expoet ules j p p 1 j p + 1 = (-1) j - p ( ) a ( j, jk ) + (-1) j - p ( ) a ( j, jk ) sice ( ) = 0 j p p 1 j p p p p p 1 p
j p p 1 p ()! p 1 p (p 1)! (j p)! = (-1) j - p [( ) + ( )] a ( j, jk ) ()! p! (j p 1)! p ()! p (p 1)! (j p)! by the distibutive popety (j p) ()! (j p) p! (j p 1)! j ()! j p! (j p)! p But ( ) + ( ) = + = + = = ( ), which is kow as Pascal s idetity [4]. This yields E p ( k, ) = (-1) j - p ( ) a ( j, jk ). j p j p VI. Coclusio I this pape, we used a ecuece elatio to build a table fo the umbe of (, ) - tiligs ad showed how diagoal ows of this table summed to odd Fiboacci umbes. We exploed combiatoial idetities based o how the two-toed tiligs wee costucted fo example, coditioig o the umbe of white tiles, the umbe of locatios of white tiles, ad the umbe of white tiles at the ed of a tilig. We elated two-toed tiligs to geealized Fiboacci umbes ad, fially, to itege compositios. We oted that compositios of coespod to (0, ) - tiligs, ad we exploed fomulas fo coutig the umbe of compositios of with at least o exactly a cetai umbe of copies of the summad k. Compositios of iteges may be used i othe fields besides mathematics fo modelig sequeces that ae subject to cetai costaits. Fo istace, a geeticist might use itege compositios to model DNA sequeces with a paticula mutatio. A compute pogamme may seek the umbe of biay sequeces of a cetai legth with oes i some miimum umbe of places. A schedule i poductio may wat to detemie the umbe of ways to split up the hous of the wokweek ito shifts of cetai legths, ad how to distibute wokes beaks so poductio pogesses smoothly. Thus we may begi to imagie how the fial thee idetities might have pactical applicatio, beyod stimulatig ou itellect. VII. Refeeces
[1] Athu T. Bejami, Phyllis Chi, Jacob N. Scott, ad Geg Simay, Combiatoics of two-toed tiligs, Fiboacci Quat. 49 (2011), o. 4, 290-297. MR2852000 [2] Athu T. Bejami ad Jeife J. Qui, Poofs that eally cout. The at of combiatoial poof, Mathematical Associatio of Ameica, Washigto, DC, 2003. MR1997773 [3] Ro Kott, Fiboacci umbes ad the golde sectio, http://www.maths.suey.ac.uk/ hosted-sites/r.kott/fiboacci/fib.html, 1996-2017 (accessed Novembe 27, 2017). [4] David R. Mazu, Combiatoics. A guided tou, Mathematical Associatio of Ameica, Washigto, DC, 2010. MR2572113