Two-Toned Tilings and Compositions of Integers
|
|
- Felicity Morgan
- 5 years ago
- Views:
Transcription
1 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
2 Cotets I. Itoductio II. Two-toed Tiligs III. Combiatoial Idetities IV. Geealizatios V. Applicatios to Compositios VI. Coclusio VII. Refeeces
3 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 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, , , , ad Note that the ode of the summads is disegaded, so is cosideed equivalet to 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 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 ad 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: , , , ad 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.
4 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).
5 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) = = 9. The a (3, 2) = 9 + 2(4) 3 = 14, ad so o Table 1: Two-toed tiligs a (, )
6 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 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 ) = = 5 = F 5. Idetity 2: Fo 0, = 0 a(, ) = F 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 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
7 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 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 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 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 t m - 2. This is idetical to the Fiboacci ecuece F m = F m 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
8 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 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 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 = 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
9 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 (, ) = ) ( ). 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
10 + 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 (, ) = 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 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 = Solvig fo v, we get v = 1 + k. By substitutio, T k = k S k = k ( D k / 2 k ) = 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 = D k = D k = 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
11 + 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 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) = = 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
12 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 j 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, 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)!! = ( ) + = ( ) + = ( ) + ( ). 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 ways. Case 2: If the fist pik tile does ot begi i cell 1, thee ae ( ) ( ) ways to 1
13 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 ( ) [( ) ( 1 )] 2-1, fom which we facto out 2-1 : [( ) 2 + ( ) ( )] = [( ) + ( )] 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 F F - k. To illustate, F 2 F 3 F 4 = + + = = 1; F 1 F 0 F 1 F 0 = + + = = 2; F 2 F 1 = + + = = 4; F 3 F 2 F 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
14 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 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 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 = 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
15 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
16 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
17 [1] Athu T. Bejami, Phyllis Chi, Jacob N. Scott, ad Geg Simay, Combiatoics of two-toed tiligs, Fiboacci Quat. 49 (2011), o. 4, MR [2] Athu T. Bejami ad Jeife J. Qui, Poofs that eally cout. The at of combiatoial poof, Mathematical Associatio of Ameica, Washigto, DC, MR [3] Ro Kott, Fiboacci umbes ad the golde sectio, hosted-sites/r.kott/fiboacci/fib.html, (accessed Novembe 27, 2017). [4] David R. Mazu, Combiatoics. A guided tou, Mathematical Associatio of Ameica, Washigto, DC, MR
The Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More information= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationMath 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as
Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationThe number of r element subsets of a set with n r elements
Popositio: is The umbe of elemet subsets of a set with elemets Poof: Each such subset aises whe we pick a fist elemet followed by a secod elemet up to a th elemet The umbe of such choices is P But this
More informationFinite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler
Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio
More informationa) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.
Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p
More informationDANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD
MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More informationGeneralized Fibonacci-Lucas Sequence
Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash
More informationConsider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample
Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More information9.7 Pascal s Formula and the Binomial Theorem
592 Chapte 9 Coutig ad Pobability Example 971 Values of 97 Pascal s Fomula ad the Biomial Theoem I m vey well acquaited, too, with mattes mathematical, I udestad equatios both the simple ad quadatical
More informationEVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS
EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationMultivector Functions
I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationSHIFTED HARMONIC SUMS OF ORDER TWO
Commu Koea Math Soc 9 0, No, pp 39 55 http://dxdoiog/03/ckms0939 SHIFTED HARMONIC SUMS OF ORDER TWO Athoy Sofo Abstact We develop a set of idetities fo Eule type sums I paticula we ivestigate poducts of
More informationUsing Counting Techniques to Determine Probabilities
Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe
More informationICS141: Discrete Mathematics for Computer Science I
Uivesity of Hawaii ICS141: Discete Mathematics fo Compute Sciece I Dept. Ifomatio & Compute Sci., Uivesity of Hawaii Ja Stelovsy based o slides by D. Bae ad D. Still Oigials by D. M. P. Fa ad D. J.L. Goss
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationSums of Involving the Harmonic Numbers and the Binomial Coefficients
Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa
More informationON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS
Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics
More information( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.
Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationAdvanced Physical Geodesy
Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig
More informationGeneralization of Horadam s Sequence
Tuish Joual of Aalysis ad Nube Theoy 6 Vol No 3-7 Available olie at http://pubssciepubco/tjat///5 Sciece ad Educatio Publishig DOI:69/tjat---5 Geealizatio of Hoada s Sequece CN Phadte * YS Valaulia Depatet
More informationSome Integral Mean Estimates for Polynomials
Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi
More informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationSome Properties of the K-Jacobsthal Lucas Sequence
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Some Popeties of the K-Jacobsthal Lucas
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationELEMENTARY AND COMPOUND EVENTS PROBABILITY
Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 ELEMENTARY AND COMPOUND EVENTS PROBABILITY William W.S. Che Depatmet of Statistics The Geoge Washigto Uivesity Washigto D.C. 003 E-mail: williamwsche@gmail.com
More informationOn randomly generated non-trivially intersecting hypergraphs
O adomly geeated o-tivially itesectig hypegaphs Balázs Patkós Submitted: May 5, 009; Accepted: Feb, 010; Published: Feb 8, 010 Mathematics Subject Classificatio: 05C65, 05D05, 05D40 Abstact We popose two
More information4. PERMUTATIONS AND COMBINATIONS
4. PERMUTATIONS AND COMBINATIONS PREVIOUS EAMCET BITS 1. The umbe of ways i which 13 gold cois ca be distibuted amog thee pesos such that each oe gets at least two gold cois is [EAMCET-000] 1) 3 ) 4 3)
More informationDisjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements
Disjoit Sets elemets { x, x, } X =, K Opeatios x Patitioed ito k sets (disjoit sets S, S,, K Fid-Set(x - etu set cotaiig x Uio(x,y - make a ew set by combiig the sets cotaiig x ad y (destoyig them S k
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationComplementary Dual Subfield Linear Codes Over Finite Fields
1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com
More informationOn a Problem of Littlewood
Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995
More informationLecture Overview. 2 Permutations and Combinations. n(n 1) (n (k 1)) = n(n 1) (n k + 1) =
COMPSCI 230: Discrete Mathematics for Computer Sciece April 8, 2019 Lecturer: Debmalya Paigrahi Lecture 22 Scribe: Kevi Su 1 Overview I this lecture, we begi studyig the fudametals of coutig discrete objects.
More informationOn the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers
O the Explicit Detemiats Sigulaities of -ciculat Left -ciculat Matices with Some Famous Numbes ZHAOLIN JIANG Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA jzh08@siacom JUAN LI Depatmet
More informationPermutations and Combinations
Massachusetts Istitute of Techology 6.042J/18.062J, Fall 02: Mathematics fo Compute Sciece Pof. Albet Meye ad D. Radhika Nagpal Couse Notes 9 Pemutatios ad Combiatios I Notes 8, we saw a vaiety of techiques
More informationCOUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS
COUNTING SUBSET SUMS OF FINITE ABELIAN GROUPS JIYOU LI AND DAQING WAN Abstact I this pape, we obtai a explicit fomula fo the umbe of zeo-sum -elemet subsets i ay fiite abelia goup 1 Itoductio Let A be
More informationSteiner Hyper Wiener Index A. Babu 1, J. Baskar Babujee 2 Department of mathematics, Anna University MIT Campus, Chennai-44, India.
Steie Hype Wiee Idex A. Babu 1, J. Baska Babujee Depatmet of mathematics, Aa Uivesity MIT Campus, Cheai-44, Idia. Abstact Fo a coected gaph G Hype Wiee Idex is defied as WW G = 1 {u,v} V(G) d u, v + d
More informationIDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks
Sémiaie Lothaigie de Combiatoie 63 (010), Aticle B63c IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks A. REGEV Abstact. Closed fomulas ae kow fo S(k,0;), the umbe of stadad Youg
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +
More informationCfE Advanced Higher Mathematics Course materials Topic 5: Binomial theorem
SCHOLAR Study Guide CfE Advaced Highe Mathematics Couse mateials Topic : Biomial theoem Authoed by: Fioa Withey Stilig High School Kae Withey Stilig High School Reviewed by: Magaet Feguso Peviously authoed
More informationDIAGONAL CHECKER-JUMPING AND EULERIAN NUMBERS FOR COLOR-SIGNED PERMUTATIONS
DIAGONAL CHECKER-JUMPING AND EULERIAN NUMBERS FOR COLOR-SIGNED PERMUTATIONS Niklas Eikse Heik Eiksso Kimmo Eiksso iklasmath.kth.se heikada.kth.se Kimmo.Eikssomdh.se Depatmet of Mathematics KTH SE-100 44
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationMath 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual
Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, 4.-4.4,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A
More informationTHE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS
THE COLORED JONES POLYNOMIAL OF CERTAIN PRETZEL KNOTS KATIE WALSH ABSTRACT. Usig the techiques of Gego Masbuam, we povide ad pove a fomula fo the Coloed Joes Polyomial of (1,l 1,2k)-petzel kots to allow
More informationKEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow
KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationLower Bounds for Cover-Free Families
Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set
More informationFibonacci and Some of His Relations Anthony Sofo School of Computer Science and Mathematics, Victoria University of Technology,Victoria, Australia.
The Mathematics Educatio ito the st Cetuy Poect Poceedigs of the Iteatioal Cofeece The Decidable ad the Udecidable i Mathematics Educatio Bo, Czech Republic, Septembe Fiboacci ad Some of His Relatios Athoy
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationA note on random minimum length spanning trees
A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationRecursion. Algorithm : Design & Analysis [3]
Recusio Algoithm : Desig & Aalysis [] I the last class Asymptotic gowth ate he Sets Ο, Ω ad Θ Complexity Class A Example: Maximum Susequece Sum Impovemet of Algoithm Compaiso of Asymptotic Behavio Aothe
More information( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to
Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special
More informationSupplementary materials. Suzuki reaction: mechanistic multiplicity versus exclusive homogeneous or exclusive heterogeneous catalysis
Geeal Pape ARKIVOC 009 (xi 85-03 Supplemetay mateials Suzui eactio: mechaistic multiplicity vesus exclusive homogeeous o exclusive heteogeeous catalysis Aa A. Kuohtia, Alexade F. Schmidt* Depatmet of Chemisty
More informationThe Stirling triangles
The Stilig tiagles Edyta Hetmaio, Babaa Smole, Roma Wituła Istitute of Mathematics Silesia Uivesity of Techology Kaszubsa, 44- Gliwice, Polad Email: edytahetmaio@polslpl,babaasmole94@gmailcom,omawitula@polslpl
More informationSection 5.1 The Basics of Counting
1 Sectio 5.1 The Basics of Coutig Combiatorics, the study of arragemets of objects, is a importat part of discrete mathematics. I this chapter, we will lear basic techiques of coutig which has a lot of
More informationON CERTAIN CLASS OF ANALYTIC FUNCTIONS
ON CERTAIN CLASS OF ANALYTIC FUNCTIONS Nailah Abdul Rahma Al Diha Mathematics Depatmet Gils College of Educatio PO Box 60 Riyadh 567 Saudi Aabia Received Febuay 005 accepted Septembe 005 Commuicated by
More informationLESSON 15: COMPOUND INTEREST
High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed
More informationEXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI
avtei EXAMPLES Solutios of AVTE by AVTE (avtei) lass XI Leade i BSE oachig 1 avtei SHORT ANSWER TYPE 1 Fid the th tem i the epasio of 1 We have T 1 1 1 1 1 1 1 1 1 1 Epad the followig (1 + ) 4 Put 1 y
More information2012 GCE A Level H2 Maths Solution Paper Let x,
GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs
More informationOn Some Fractional Integral Operators Involving Generalized Gauss Hypergeometric Functions
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 5, Issue (Decembe ), pp. 3 33 (Peviously, Vol. 5, Issue, pp. 48 47) Applicatios ad Applied Mathematics: A Iteatioal Joual (AAM) O
More informationCSE 191, Class Note 05: Counting Methods Computer Sci & Eng Dept SUNY Buffalo
Coutig Methods CSE 191, Class Note 05: Coutig Methods Computer Sci & Eg Dept SUNY Buffalo c Xi He (Uiversity at Buffalo CSE 191 Discrete Structures 1 / 48 Need for Coutig The problem of coutig the umber
More informationOn the Combinatorics of Rooted Binary Phylogenetic Trees
O the Combiatoics of Rooted Biay Phylogeetic Tees Yu S. Sog Apil 3, 2003 AMS Subject Classificatio: 05C05, 92D15 Abstact We study subtee-pue-ad-egaft (SPR) opeatios o leaf-labelled ooted biay tees, also
More informationNew Sharp Lower Bounds for the First Zagreb Index
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A:APPL. MATH. INFORM. AND MECH. vol. 8, 1 (016), 11-19. New Shap Lowe Bouds fo the Fist Zageb Idex T. Masou, M. A. Rostami, E. Suesh,
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationRECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia.
#A39 INTEGERS () RECIPROCAL POWER SUMS Athoy Sofo Victoia Uivesity, Melboue City, Austalia. athoy.sofo@vu.edu.au Received: /8/, Acceted: 6//, Published: 6/5/ Abstact I this ae we give a alteative oof ad
More informationCombinatorially Thinking
Combiatorially Thiig SIMUW 2008: July 4 25 Jeifer J Qui jjqui@uwashigtoedu Philosophy We wat to costruct our mathematical uderstadig To this ed, our goal is to situate our problems i cocrete coutig cotexts
More information4. PERMUTATIONS AND COMBINATIONS Quick Review
4 ERMUTATIONS AND COMBINATIONS Quick Review A aagemet that ca be fomed by takig some o all of a fiite set of thigs (o objects) is called a emutatio A emutatio is said to be a liea emutatio if the objects
More information1 Counting and Stirling Numbers
1 Coutig ad Stirlig Numbers Natural Numbers: We let N {0, 1, 2,...} deote the set of atural umbers. []: For N we let [] {1, 2,..., }. Sym: For a set X we let Sym(X) deote the set of bijectios from X to
More informationRELIABILITY ASSESSMENT OF SYSTEMS WITH PERIODIC MAINTENANCE UNDER RARE FAILURES OF ITS ELEMENTS
Y Geis ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS ELEMENTS T&A # (6) (Vol) 2, Mach ELIABILITY ASSESSMENT OF SYSTEMS WITH PEIODIC MAINTENANCE UNDE AE FAILUES OF ITS
More informationOn the maximum of r-stirling numbers
Advaces i Applied Mathematics 4 2008) 293 306 www.elsevie.com/locate/yaama O the maximum of -Stilig umbes Istvá Mező Depatmet of Algeba ad Numbe Theoy, Istitute of Mathematics, Uivesity of Debece, Hugay
More informationsubcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016
subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may
More informationThe Multivariate-t distribution and the Simes Inequality. Abstract. Sarkar (1998) showed that certain positively dependent (MTP 2 ) random variables
The Multivaiate-t distibutio ad the Simes Iequality by Hey W. Block 1, Saat K. Saka 2, Thomas H. Savits 1 ad Jie Wag 3 Uivesity of ittsbugh 1,Temple Uivesity 2,Gad Valley State Uivesity 3 Abstact. Saka
More informationA NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS
Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com
More informationDiscussion 02 Solutions
STAT 400 Discussio 0 Solutios Spig 08. ~.5 ~.6 At the begiig of a cetai study of a goup of pesos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the fiveyea study, it was detemied
More informationOn composite conformal mapping of an annulus to a plane with two holes
O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy
More informationStrong Result for Level Crossings of Random Polynomials
IOSR Joual of haacy ad Biological Scieces (IOSR-JBS) e-issn:78-8, p-issn:19-7676 Volue 11, Issue Ve III (ay - Ju16), 1-18 wwwiosjoualsog Stog Result fo Level Cossigs of Rado olyoials 1 DKisha, AK asigh
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More informationResearch Article The Peak of Noncentral Stirling Numbers of the First Kind
Iteatioal Joual of Mathematics ad Mathematical Scieces Volume 205, Aticle ID 98282, 7 pages http://dx.doi.og/0.55/205/98282 Reseach Aticle The Peak of Nocetal Stilig Numbes of the Fist Kid Robeto B. Cocio,
More informationChapter 2 Sampling distribution
[ 05 STAT] Chapte Samplig distibutio. The Paamete ad the Statistic Whe we have collected the data, we have a whole set of umbes o desciptios witte dow o a pape o stoed o a compute file. We ty to summaize
More informationTaylor Transformations into G 2
Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that
More informationSolutions to Final Exam
Solutios to Fial Exam 1. Three married couples are seated together at the couter at Moty s Blue Plate Dier, occupyig six cosecutive seats. How may arragemets are there with o wife sittig ext to her ow
More informationLet us consider the following problem to warm up towards a more general statement.
Lecture 4: Sequeces with repetitios, distributig idetical objects amog distict parties, the biomial theorem, ad some properties of biomial coefficiets Refereces: Relevat parts of chapter 15 of the Math
More information