MATH 681 Notes Combinatorics and Graph Theory I. ( 4) n. This will actually turn out to be marvelously simplifiable: C n = 2 ( 4) n n + 1. ) (n + 1)!

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1 MATH 681 Nots Combiatorics ad Graph Thory I 1 Catala umbrs Prviously, w usd gratig fuctios to discovr th closd form C = ( 1/ +1) ( 4). This will actually tur out to b marvlously simplifiabl: ( ) 1/ C = ( 4) + 1 = ( 4) ( )( )( ) ( 1 ) ( + 1)! ( 1)( 3)( 5) (1 ) = ( 4) ( + 1)! ( 1) = ( + 1)! ( 1)() = 4 6 ( )()( + 1)! = ()! =!( + 1)! ()!!!( + 1) = A purly combiatorial approach ( ) If w d rathr ot hav this much algbra, tak hart: thr s a simplr, lgat, combiatorial argumt. Lt us itrprt C as th umbr of paths with up-stps ad dow-stps which do t go blow thir startig poit (such a path is calld a Dyck path. Lt us cosidr such a path with = 4, draw with th y-axis rprstig hight ad th x-axis rprstig tim: If w rotat this 45 dgrs coutrclockwis, ad impos a grid structur, w s that this structur corrspods to a combiatorial objct much lik som w v coutd bfor: So w might ot that C 4 is xactly th umbr of 8-stp walks o th followig grid: Pag 1 of 7 Octobr 15, 009

2 MATH 681 Nots Combiatorics ad Graph Thory I W could do this usig iclusio-xclusio umratio of walks through (1, 0), (, 1), (3, ) ad (4, 3), but ufortuatly, thr ar walks goig through ach of thos i vry combiatio, ach of thm diffrt i umbr, so such a iclusio-xclusio would b a hidous 16-trm mostrosity! Fortuatly, thr is a simpl way to cout ths aftr all, by buildig a bijctio btw walks to (, ) which touch th subdiagoal ad walks to ( + 1, 1). Cosidr a walk from (0, 0) to (, ) which is ot always abov th subdiagoal. Th, thr is at last o poit (x, y) o it whr x > y; cosidr th first such poit, which ca asily b s to hav th form (x, x 1). W may cosidr th walk from (0, 0) to (, ) as big a compositio of two walks: o from (0, 0) to (x, x 1), ad o from (x, x 1) to (, ). This lattr walk cosists of x right-stps ad x + 1 up-stps. Now lt us flip this walk, rplacig up-stps with right-stps ad vic-vrsa, so ow w hav a walk from (x, x 1) cosistig of x + 1 right-stps ad x up-stps; this will b a walk to ( + 1, 1). Thus, w may map ach walk from (0, 0) to (, ) which gos blow th diagoal to a walk from (0, 0) to (h + 1, 1). To dmostrat that this is a bijctio, lt us obsrv how this procss could b rvrsd. A walk from (0, 0) to (+1, 1) must cssarily pass through a poit (x, y) whr x > y: th dstiatio is such a poit, v if o othr poit is. Cosidr th first such poit i th walk; as abov, its coordiats must b (x, x 1), ad ow w cosidr this walk as a compositio of a walk from (0, 0) to (x, x 1) ad from (x, x 1) to ( + 1, 1). Th lattr walk cosists of x + 1 right-stps ad x up-stps; flippig it givs us a walk from (0, 0) to (, ) passig through (x, x 1). Thus, to fid th umbr of walks which do ot dip blow th diagoal i a grid, w tak th umbr of walks i a grid, ad subtract th ubmr of walks i a ( + 1) ( 1) grid. This is asily calculatd to b ( ) ( 1 Stirlig s approximatio ) =!!!! ( + 1)!( 1)! =!!! ( 1 ) = ( ) Coutig th umbr of ways somthig ca happ has som barig o computatioal complxity. If a computr was, for istac, istructd to fid th pair of umbrs with last diffrc i a st o ubmrs, o way would b to work by brut forc, tstig vry pair of umbrs, which would ivolv ( ) idividual tsts: this algorithm would tak ( 1) = stps, which would b calld, i algorithmic circls, simply a O( ) algorithm to rprst th asymptotically largst trm i th umbr of stps, without a cofficit. So havig a ida of th rough asymptotic valu of, for istac, a biomial cofficit, ca b quit usful. W ca s asily that ( ) = O( ) ad that ( ) = O( 3 ), but w ar o shakir groud tryig to say that i gral ( ) k k k! = O( k ), sic if k is larg ough, th trms i th umrator of ( 1) ( k+1) k! ar ot actually all that clos to. A tool which srvs us wll i approximatig biomials (amog othr thigs) is calld Stirlig s approximatio. Thorm 1. For sufficitly larg,! π ( ) ; mor prcisly, it is tru that lim! π ( ) = 1 Pag of 7 Octobr 15, 009

3 MATH 681 Nots Combiatorics ad Graph Thory I This formula is traditioally prov usig aalytical mthods, somtims ivokig th dfiitio of th gamma fuctio:! = Γ( + 1) = 0 t t dt Th aalytical mthods usd to show that this is qual to π ( ) Ψ(), whr Ψ() tds towards 1 for larg, ar rathr byod th scop of this cours; a dct proof appars i Rudi s Pricipls of Mathmatical Aalysis. Howvr, our mai us for Stirlig s approximatio will b i givig altrativ, factorial-fr forms for our favorit umratios. For istac, w ca dtrmi, asymptotically, what ( k) is roughly qual to, i trms of ad k: ( )! = k ( k)!k! ( π ) ( π( k) k ) k ( πk k ) k π( k)k ( k) k k k 1 +1/ π ( k) k+1/ k k+1/ Wh k, w ca assum k ad trat k k+1/ as a arbitrary costat, so th abov would bcom: ( ) 1 +1/ k πk k+1/ k+1/ = k πk k+1/ so that if k is a costat small with rspct to, w ca justifiably claim that ( k) = O( k ). But if k is a sigificat proportio of, for istac, if k = c for som costat 0 < c < 1: ( ) c 1 π +1/ ((1 c)) (1 c)+1/ (c) c+1/ 1 πc(1 c) [(1 c) 1 c c c ] which would asymptotically b O( a ), whr a = 1 (1 c) 1 c c c. Turig to a spcific usful xampl whr th trms of biomials ar costat multipls of ach othr, w may cosidr drivig a approximatio for th abov-dtrmid formula for th Catala umbrs: C = ( ) 1 ( + 1) () +1/ π +1/ +1/ 1 ( + 1) π 4 ( + 1) π +1/ 1/ Pag 3 of 7 Octobr 15, 009

4 MATH 681 Nots Combiatorics ad Graph Thory I This is a vry simpl formula, ad o that givs a obvious asymptotic of O( 4 3/. Ad it s actually fairly accurat! C (+1) π Icidtally, basd o our prvious formula C = ( 1/ +1) ( 4), w ow hav a strog suggstio that ( ) 1/ +1 ( 1). This is t trribly usful o its ow, but it s cut. (+1) π 3 Symmtry ad th Polya Mthod W flirtd with symmtry-rductio way back at th bgiig of th cours. Lt s look at a pair of sampl qustios illustratig typs of symmtry ad how w d dal with it: Qustio 1: W hav 7 distict popl from which w d lik to form a bridg tabl (of 4 popl i a ordr). Rotatios of th sam foursom ar idtical. How may foursoms ar thr? Aswr 1: If w just cosidr a ordrd list of 4 popl, this is a simpl umratio statistic: 7P 4 = 7! 3! = 840. But this icluds th sam foursoms multipl tims: ot that ABCD is th sam as DABC, CDAB, ad BCDA (but ot, for istac, ADCB or BACD, which ar ot simpl rotatios). Fortuatly, vry foursom will b rprstd xactly four tims, sic it ad its 3 rotatios ar guaratd to appar i our list of 840. Thus, thr ar = 10 bridg foursoms. Qustio : W hav 7 idividual distict bads, ad would lik to put 4 o a cklac. Rotatios ad flips of a sigl cklac ar cosidrd to b idtical. How may diffrt cklacs ar possibl? Aswr : This is much lik abov, but hr, istad of havig oly 4 cklacs quivalt to ABCD, w hav 8: ABCD itslf, DABC, CDAB, BCDA, DCBA, ADCB, BADC, ad CBAD. Thus thr ar = 105 cklacs. Ths ar simpl ways of wiowig dow symmtris. Lt s thik of a mor difficult cas! Qustio 3: W hav a ulimitd supply of black ad whit bads. W wat to mak a 5-bad cklac. How may diffrt ways ar thr to do this? Aswr 3: This is ot asily do with pur symmtry divisio. W might thik that thr ar 10 trasformatios, so ach cklac is rprstd 10 tims if w wr to just cosidr straight strigs, but thr would b 5 = 3 strigs, ad 5 10 = 3. is, first of all, ot a itgr, ad scod, too small. If w wr to brut-forc this, w ll fid 8 cklacs (all-whit, o black, two adjact blacks, two o-adjact blacks, two adjact whits, two o-adjact whits, o whit, all-black). How could w possibly gt this rsult i a ssibl mar? To do this, w r goig to d to discuss quivalc classs ad prmutatio groups. 3.1 Prmutatio groups W v coutrd prmutatios as combiatorial objcts, but hr w r goig to wat to discuss thir algbraic proprtis. Pag 4 of 7 Octobr 15, 009

5 MATH 681 Nots Combiatorics ad Graph Thory I Dfiitio 1. A prmutatio of lgth is a bijctiv fuctio π : {1,, 3,..., } {1,, 3,..., }. Th product πσ of two prmutatios π ad σ is thir compositio as fuctios, π σ. Th ivrs π 1 of a prmutatio π is its ivrs as a fuctio. Not that prmutatio products ar ot commutativ i gral: πσ is likly to b a diffrt prmutatio tha σπ. I additio, for big th idtity prmutatio (i.. th idtity fuctio o {1,, 3,..., }, it is asy to s that π = π = π, ad that ππ 1 = π 1 π =. Togthr with th associativ law of multiplicatio, ths proprtis guarat that th prmutatios form th algbraic structur kow as a group; this group is kow as S, th symmtric group, ad w kow, from our combiatorial xploratios, that S =!. Thr ar two stadards for rprstig a prmutatio: thr is mappig otatio, i which π(1), π(), π(3), tc. ar listd i ordr; thr is also cycl otatio, i which o lists ach lmt followd by its imag, rpatig util th cycl rturs to th bgiig. For istac, cosidrig a lmt of C 6 with π(1) = 3, π() =, π(3) = 5, π(4) = 6, π(5) = 1, ad π(6) = 4, this prmutatio could b rprstd i mappig otatio as (35614) ad i cycl otatio as (135)()(46). I ths ots, mappig otatio is usd for brvity. Ay subgroup of S is kow as a symmtry group. From a algbraically aïv prspctiv, w ca dscrib a symmtry group as such: Dfiitio. A symmtry group G is a ompty st of prmutatios of th sam lgth such that: 1. th idtity is i G.. for σ ad π i G, σπ is i G. 3. for π i G, π 1 is i G. As it turs out, oly th scod of ths coditios is actually cssary (th first follows sic for ay π i G, thr is a N such that π N =, ad th third follows by otig that π N 1 = π 1 ), but th first ad third giv a bttr ida of its structur. Thr ar a trmdous varity of symmtry groups, ad th rsults w will s latr apply to all of thm. For ow, though, w will focus o thos with a gomtric rprstatio. 3. -dimsioal symmtris Symmtris i th -dimsioal pla ar mappigs of a rgular -go oto itslf. W ca visio two such symmtris: rotatioal oly (such as i our bridg gam xampl) or rotatio with rflctio (as i our cklac xampl). If oly rotatios ar allowd, w ca map a -go to itslf with a rotatio of πk radias; that is to say, with ay multipl of 1 rvolutios. This yilds diffrt possibl prmutatios: th idtity rotatio of 0 radias, a rotatio of π radias, a rotatio of 4π radias, ad so forth up to π( 1) radias. Th xt rotatio i this squc would b idtical to th idtity, ad thus has alrady b coutd. Algbraically, ths ca b rprstd as forward-shifts of th vrtics: if w labl th vrtics of a -go cyclically, th ths rotatios would rspctivly b = (13 ), r = (34 1), r = (34 1), ad so forth up to r 1 = (13 ( )( 1)). Pag 5 of 7 Octobr 15, 009

6 MATH 681 Nots Combiatorics ad Graph Thory I This group is associatd with th cyclic group, dotd C or Z, ad ca b thought of i trms of modular arithmtic: r a + r b = r c, whr c a + b (mod ). If w allow rflctio, th w gt a largr group, which icluds all th rotatios, ad i additio, icluds othr prmutatios. If w idtify ths othr prmutatios algbraically, w ca dfi f = (( 1)( ) 31) ad th ot that a group cotaiig f ad r must also cotai fr = (( 1)( ) 31), fr = (( )( 3) 1( 1)) ad so forth up to fr 1 = (1( 1) 43). Altrativly, gomtrically, w ca ot that a compositio of rflctios with rotatios is itslf a rflctio, ad that thr ar rotatios mappig a -go oto itslf. Th axs of rotatio look somwhat diffrt dpdig o whthr is v or odd, but thr ar always of thm: This prmutatio group is kow as th dihdral group D. D =, ad covtioally, lmts of D ar writt as fr k with 0 k <. Th product of two lmts is dtrmid accordig to th ruls r = 1, f = 1, ad fr = r 1 f dimsioal symmtris Symmtry i 3-spac is grally limitd to th 5 platoic solids, which mas that thr ar 3 figurs i 3-spac whos symmtris ar frqutly xplord: th ttrahdro, th cub (which has th sam symmtris as th octahdro, which is dual to th cub), ad th icosohdro (which has th sam symmtris as th dodcahdro) Rotatioal axs which map a solid to itslf must pass through opposit faturs of th solid: ithr through th ctrs of a pair of opposit facs, a pair of opposit vrtics, th midpoits of a pair of opposit dgs, or th ctr of a fac opposit a vrtx. This allows us to umrat th rotatioal axs of ay solid with comparativ as. W may start by cosidrig a ttrahdro. A ttrahdro has 4 vrtics ad 4 facs, with facs opposit vrtics, ad 6 dgs. W thus hav 7 prospctiv axs of symmtry: 4 draw btw a vrtx ad its opposit fac, ad 3 draw btw opposit pairs of dgs. Now w may us ths axs of symmtry to idtify all th prmutatios corrspodig to rotatios of th ttrahdro. Thr is th idtity prmutatio, which corrspods to a zro rotatio. For ach of th vrtx-to-fac axs, a viw of th ttrahdro from abov has triagular symmtry, so w could rotat aroud ths axs ithr 10 or 40 ; w hav 4 such axs with possibl rotatios, givig ight prmutatios. Lastly, for ach of th thr dg-to-dg axs, w may rotat th dg oto itslf via a 180 rotatio, givig thr mor prmutatios. Thus, thr ar = 1 rotatios mappig th ttrahdro oto itslf. This group is somtims calld th chiral ttrahdral symmtry group, but is also associatd with th algbraic group calld th altratig group of dgr 4, or A 4. If w add rflctios to this group, or rathr, add a sigl rflctio ad cout o compositio with rotatios to yild othr rflctios, th w doubl th group s siz, gttig 4 automorphisms of Pag 6 of 7 Octobr 15, 009

7 MATH 681 Nots Combiatorics ad Graph Thory I th ttrahdro. This is th achiral ttrahdral symmtry group, algbraically associatd with th group of all prmutatios o 4 lmts, th symmtric group S 4. Movig o w may look at th cub: it has 6 facs dividd ito 3 opposit pairs, 1 dgs dividd ito 6 opposit pairs, ad 8 vrtics dividd ito 4 opposit pairs. Thus thr ar 13 possibl axs of rotatio through th cub. Lookig dirctly at a fac, th cub has 4-fold symmtry. Thus, th fac-to-fac axs ca b rotatd 90, 180, or 70 to map th cub back oto itslf. Sic thr ar 3 such axs ad 3 possibl rotatios about ach, ths yild 9 prmutatios of th cub. Th dg-to-dg axs ca oly map th cub to itslf by goig through a 180 rotatio, so ths 6 axs cotribut o prmutatio ach for a total of 6. Lastly, viwig a corr of th cub, th cub has thrfold symmtry, so a vrtx-to-vrtx axis ca b rotatd ithr 10 or 40 ; w hav 4 such axs with possibl rotatios, givig 8 prmutatios. Togthr with th idtity prmutatio, th abov total = 4 i umbr. This group is th chiral octahdral symmtry group, dotd O, ad somtims cosidrd as a subgroup of S 6. It is i fact isomorphic to th achiral ttrahdral symmtry group, although it is rarly cosidrd i that cotxt. As abov, addig a rflctio doubls th ordr of th group, so th automorphisms of th cub by rotatio ad rflctio umbr 48; this group is calld th achiral octahdral symmtry group or simply th octahdral group ad is dotd O h. Pag 7 of 7 Octobr 15, 009