O Dtrmiistic Fiit Automata ad Sytactic Mooid Siz, Cotiud Markus Holzr ad Barbara Köig Istitut für Iformatik, Tchisch Uivrsität Müch, Boltzmastraß 3, D-85748 Garchig bi Müch, Grmay mail: {holzr,koigb}@iformatik.tu-much.d Abstract. W cotiu our ivstigatio o th rlatioship btw rgular laguags ad sytactic mooid siz. I this papr w cofirm th cojctur o two grator trasformatio smigroups. W show that for vry prim 7 thr xist atural umbrs k ad l with = k + l such that th smigroup U k,l is maximal w.r.t. its siz amog all (trasformatio) smigroups which ca b gratd with two grators. This sigificatly tights th boud o th sytactic mooid siz of laguags accptd by -stat dtrmiistic fiit automata with biary iput alphabt. As a by-product of our ivstigatios w ar abl to dtrmi th maximal siz amog all smigroups gratd by two trasformatios, whr o is a prmutatio with a sigl cycl ad th othr is a o-bijctiv mappig. 1 Itroductio Fiit automata ar usd i svral applicatios ad implmtatios i softwar girig, programmig laguags ad othr practical aras i computr scic. Thy ar o of th first ad most itsly ivstigatd computatioal modls. Sic rgular laguags hav may rprstatios i th world of fiit automata it is atural to ivstigat th succictss of thir diffrt rprstatios. Rctly, th siz of th sytactic mooid as a atural masur of dscriptiv complxity for rgular laguags was proposd i [3] ad studid i dtail. Rcall, that th sytactic mooid of a laguag L is th smallst mooid rcogizig th laguag udr cosidratio. It is uiquly dfid up to isomorphism ad is iducd by th sytactic cogruc L dfid ovr Σ by v 1 L v 2 if ad oly if for vry u, w Σ w hav uv 1 w L uv 2 w L. Th sytactic mooid of L is th quotit mooid M(L) = Σ / L. I particular, th siz of trasformatio mooids of -stat (miimal) dtrmiistic fiit automata was ivstigatd i [3]. I most cass tight uppr bouds o th sytactic mooid siz wr obtaid. It was prov that a - stat dtrmiistic fiit automato with siglto iput alphabt (iput alphabt with at last thr lttrs, rspctivly) iducs a liar (, rspctivly) siz sytactic mooid. I th cas of two lttr iput alphabt, a lowr boud of ( ) l l! k ( l) k k l l, for som atural umbrs k ad l clos to 2, ad a trivial
o-matchig uppr boud of!+g(), whr g() dots Ladau s fuctio [5 7], which givs th maximal ordr of all prmutatios i S, for th siz of th sytactic mooid of a laguag accptd by a -stat dtrmiistic fiit automato was giv. This iducs a family of dtrmiistic fiit automata such that th fractio of th siz of th iducd sytactic mooid ad tds to 1 as gos to ifiity, ad is th startig poit of our ivstigatios. I this papr w tight th boud o th sytactic mooid siz o two grators, cofirmig th cojctur, that for vry prim 7 thr xist atural umbrs k ad l with = k + l such that th smigroup U k,l as itroducd i [3] is maximal w.r.t. its siz amog all (trasformatio) smigroups which ca b gratd with two grators. Sic U k,l, for suitabl k ad l is a sytactic mooid, this sharps th abov giv boud for sytactic mooids iducd by -stat dtrmiistic fiit automata with biary iput alphabt. I ordr to show that thr is o largr subsmigroup of T with two grators, w ivstigat all possibl combiatios of grators. I pricipl th followig situatios for grators appar: 1. Two prmutatios, 2. a prmutatio with o cycl ad a o-bijctiv trasformatio, 3. a prmutatio with two or mor cycls ad a o-bijctiv trasformatio th smigroup U k,l is of this typ, ad 4. two o-bijctiv trasformatios. I th forthcomig w will show that for a larg ough th maximal subsmigroup is of typ (3) ad that whvr is prim th smigroup is isomorphic to som U k,l. Th tir argumt rlis o a sris of lmmata covrig th abov mtiod cass, whr th scod cas plays a major rol. I fact, as a by-product w ar abl to dtrmi th maximal siz amog all smigroups gratd by two trasformatios, whr o trasformatio is a prmutatio with a sigl cycl ad th othr is a o-bijctiv mappig. I ordr to achiv our goal w us divrs tchiqus from algbra, aalysis, ad v computr vrifid rsults for a fiit umbr of cass. Th papr is orgaizd as follows. I th xt sctio w itroduc th cssary otatios. Th i Sctio 3 w start our ivstigatios with th cas whr o is a prmutatio with a sigl cycl ad th othr is a o-bijctiv mappig. Nxt, two prmutatios ad two o-bijctiv mappigs ar cosidrd. Sctio 5 dals with th most complicatd cas, whr th prmutatio cotais two or mor cycls, ad Sctio 6 is dvotd to th mai rsult of this papr, o th siz maximality of th smigroup udr cosidratio. Fially, w summariz our rsults ad stat som op problms. 2 Dfiitios W assum th radr to b familiar with th basic otios of formal laguag thory ad smigroup thory, as cotaid i [4] ad [9]. I this papr w ar dalig with rgular laguags ad thir sytactic mooids. A smigroup is a
o-mpty st S quippd with a associativ biary opratio, i.., (αβ)γ = α(βγ) for all α, β, γ S. Th smigroup S is calld a mooid if it cotais a idtity lmt id. If E is a st, th w dot by T (E) th mooid of fuctios from E ito E togthr with th compositio of fuctios. W rad compositio from lft to right, i.., first α, th β. Bcaus of this covtio, it is atural to writ th argumt i of a fuctio to th lft: (i)αβ = ((i)α)β. Th imag of a fuctio α i T (E) is dfid as img(α) = { (i)α i E } ad th krl of α is th quivalc rlatio, which is iducd by i j if ad oly if (i)α = (j)α. I particular, if E = {1,..., }, w simply writ T for th mooid T (E). Th mooid of all prmutatios ovr lmts is dotd by S ad trivially is a sub-mooid of T. Th smigroup w ar itrstd i is dfid blow ad was itroducd i [3] i ordr to study th rlatio btw -stat dtrmiistic fiit automata with biary iput alphabt ad th siz of sytactic mooids. Dfiitio 1. Lt 2 such that = k + l for som atural umbrs k ad l. Furthrmor, lt α = (1 2... k)(k + 1 k + 2... ) b a prmutatio of S cosistig of two cycls. W dfi th smigroup U k,l as a subst of T as follows: A trasformatio γ is a lmt of U k,l if ad oly if 1. thr xists a atural umbr m N such that γ = α m or 2. th trasformatio γ satisfis that (a) thr xist i {1,..., k} ad j {k + 1,..., } such that (i)γ = (j)γ ad (b) thr xists h {k + 1,..., } such that h img(γ). Obsrv, that it is always bttr to choos th lmt h which is missig i th imag of γ from th largr cycl of α sic this yilds a largr smigroup U k,l. Thrfor w ca safly assum that k l. I [3] it was show that if gcd{k, l} = 1, th th smigroup U k,l ca b gratd by two grators oly. Morovr, i this cas, U k,l is th sytactic mooid of a laguag accptd by a -stat dtrmiistic fiit automato, whr = k + l. Fially, w d som additioal otatio. If A is a arbitrary o-mpty subst of a smigroup S, th th family of subsmigroups of S cotaiig A is o-mpty, sic S itslf is o such smigroup; hc th itrsctio of th family is a subsmigroup of S cotaiig A. W dot it by A. It is charactrizd withi th st of subsmigroups of S by th proprtis: (1) A A ad (2) if U is a subsmigroup of S cotaiig A, th A U. Th smigroup A cosists of all lmts of S that ca b xprssd as fiit products of lmts i A. If A = S, th w say that A is a st of grators for S. If A = {α, β} w simply writ A as α, β. 3 Smigroup Siz Th Sigl Cycl Cas I this sctio w cosidr th cas whr o grator is a prmutatio cotaiig a sigl cycl ad th othr is a o-bijctiv trasformatio. This situatio
is of particular itrst, sic it allows us to compltly charactriz this cas ad morovr it is vry hlpful i th squl wh dalig with two prmutatios or two o-bijctiv trasformatios. Th outli of this sctio is as follows: First w dfi a subst of T by som asy proprtis as i th cas of th U k,l smigroup, vrify that it is a smigroup ad that it is gratd by two grators. Th subst of T w ar itrstd i, is dfid as follows: Dfiitio 2. Lt 2 ad 1 d <. Furthrmor, lt α = (1 2 3... ) b a prmutatio of S cosistig of o cycl. W dfi V d as a subst of T as follows: A trasformatio γ is a lmt of V d if ad oly if 1. thr xists a atural umbr m N such that γ = α m or 2. thr xists a i {1,..., } such that (i)γ = (i + d)γ, whr + dots th additio modulo. Th ituitio bhid choosig this spcific smigroup V d is th followig: Without loss of grality w ca assum that α = (1 2 3... ). By choosig a o-bijctiv trasformatio β which maps two lmts 1 i < j oto th sam imag o ca ifr that vry trasformatio γ gratd by α ad β is ithr a multipl of α or maps two lmts of distac d := j i to th sam valu. Nxt w show that V d is idd a smigroup ad that V d is isomorphic to V d if gcd{, d} = gcd{, d }. Thrfor, it will b sufficit to cosidr oly divisors of i th followig. W omit th proof of th followig lmma. Lmma 1. Th st V d is closd udr compositio ad is thrfor a (trasformatio) smigroup. Morovr, V d is isomorphic to V d whvr d = gcd{, d }. Bfor w ca prov that V d ca b gratd by two lmts of T w d a rsult, which costituts how to fid a complt basis for th symmtric group S. Th rsult giv blow was show i [8]. Thorm 1. Giv a o-idtical lmt α i S, th thr xists β such both grat th symmtric group S, providd that it is ot th cas that = 4 ad α is o of th thr prmutatios (1 2)(3 4), (1 3)(2 4), ad (1 4)(2 3). Now w ar rady for th proof that two lmts ar ough to grat all of th smigroup V d. Du to th lack of spac w omit th proof of th followig thorm, which is havily basd o Thorm 1. Thorm 2. Lt 2 ad 1 d <. Th smigroup V d ca b gratd by two lmts of T, whr o lmt is th prmutatio α = (1 2 3... ) ad th othr is a lmt β of krl siz 1. 1 I ordr to dtrmi th siz of V d, th followig thorm, rlatig siz ad umbr of colourigs of a particular graph, is vry usful i th squl. 1 Obsrv, that thr is a -stat miimal dtrmiistic fiit automato A with biary iput alphabt th trasitio mooid of which quals V d. Hc, V d is th sytactic mooid of L(A). Sic this statmt ca b asily s, w omit its proof.
Thorm 3. Lt 2 ad 1 d < with d. Dot th udirctd graph cosistig of d circls, ach of lgth d, by G. Th V d = + N, whr N = ( ( 1) d + ( 1) d ( 1) ) d is th umbr of ivalid colourigs of G with colours. Proof. Th subsmigroup V d ca b obtaid from T by rmovig all trasformatios ot satisfyig th scod part of Dfiitio 2 ad by addig th multipls of α aftrwards. Th umbr of th formr trasformatios ca b dtrmid as follows: Assum that a graph G has ods V = {1,..., } whr a circl C k cosists of ods {k, k + d,..., k + id,..., k + d}, for 1 k d. Th o ca asily vrify that th colourigs of G ar xactly th trasformatios which do ot satisfy th scod part of Dfiitio 2. Th umbr of colourigs of a graph G with k colours is dscribd by its chromatic polyomial, s,.g. [10]. Sic th chromatic polyomial of a circl C with ods is (k 1) + ( 1) (k 1) ad th chromatic polyomial of a graph cosistig of discoctd compots is th product of th chromatic polyomials of its compots, th dsird rsult follows. Now w ar rady to prov som asymptotics o th siz of V d for som particular valus of d, which ar dtrmid first. Thorm 4. Th siz of V d is maximal whvr d = max({1} { d d divids ad d is odd }). Lt V dot th smigroup V d of maximal siz. Th V lim = 1 1, whr is th bas of th atural logarithm. Proof. Th maximality of V d w.r.t. its siz is s as follows. W first dfi two ral-valud fuctios ( ) x ( ) x u v,k (x) = ( 1) k x + ( 1) ad u odd,k (x) = ( 1) k x ( 1). Th additioal idx k is prst for latr us s Lmma 5. For ow w assum that k =. W hav V d = + u v, (d) whvr d is v ad V d = + u odd,(d) whvr d is odd. Obviously uodd,k < uv,k. First w show that uv,k is strictly mooto by takig th first drivatio of l u v,k (x). W obtai d ) l uv,k (x) = l (( 1) k x + ( 1) + x ( 1) k x l( 1) ( ) k x 2 dx ( 1) k x + ( 1) ( ) > l ( 1) k x k ( 1) k x l( 1) x ( 1) k x = k x l( 1) k l( 1) = 0 x
Aalogously o ca show that u odd,k is strictly atito. So if thr xist divisors d such that d is odd, th smigroup V d is maximal w.r.t. its siz whvr w choos th largst such d. Othrwis thr ar oly divisors d such that d is v ad w choos th smallst of ths divisors which is 1. Nxt cosidr th smigroup V = V d, for som 1 d <. From our prvious ivstigatios o ca ifr that th followig iqualitis hold: + ( 1) ( 1) + ( ( 1) d + ( 1) d ( 1) ) d + ( ( 1) 3 ( 1) ) 3. Th scod half of th iquality follows sic th siz of V d is maximal whvr d is odd ad 1 d < is maximal. This is achivd idally whvr d = 3. Th rst follows with th mootoicity ad atitoicity of th fuctios u v, ad u odd,, rspctivly. W ow dtrmi th limits of th lowr ad uppr bouds. Thr w fid that + ( 1) ( ( 1) lim = lim 1 + 1 = 1 lim = 1 1, ( ) ) 1 ( 1 ) 1 1 lim sic lim (1 + 1 ) =, ad th limit of th uppr boud tds also to 1 1 by similar rasos as abov. Hc lim V = 1 1. From th asymptotic bhaviour of th smigroups V ad U k,l w immdiatly ifr th followig thorm. Thorm 5. Thr xists a atural umbr N such that for vry N, thr xist k ad l with = k + l such that V < U k,l. Proof. Th xistc of a atural umbr N satisfyig th rquirmts giv abov follows from Thorm 4 ad a rsult from [3], which stat that V lim = 1 1 for suitabl k() ad l(). ad U k(),l() lim = 1, Th followig lmma shows that whvr w hav a prmutatio cosistig of a sigl cycl ad a o-bijctiv trasformatio, w obtai at most as may lmts as cotaid i V. Lmma 2 (A cycl ad a o-bijctiv trasformatio). If α S such that α cosists of a sigl cycl ad β T \S, th α, β V.
Proof. Sic th prmutatio α cosists of a sigl cycl, thr is a prmutatio π such that παπ 1 = (1 2 3... ). W st α = παπ 1 ad β = πβπ 1. Bcaus π is a bijctio, w ca ifr that α, β = α, β. Thr ar two lmts i < j such that (i)β = (j)β. W dfi d = j i. It ca b asily s that α ad β grat at most th trasformatios spcifid i Dfiitio 2. Thrfor w coclud that α, β V. Obsrv, that bcaus of Thorm 5, Lmma 2 implis that thr xists a atural umbr N such that for vry N thr xist k ad l with = k + l such that α, β < U k,l, for vry α S such that α cosists of a sigl cycl ad β T \S. 4 Smigroup Siz Two Prmutatios or No-Bijctiv Mappigs I this sctio w show that two prmutatios or two o-bijctiv trasformatio ar ifrior i siz to a U k,l smigroup, for larg ough = k + l. Hr it turs out, that th smigroup V is vry hlpful i both cass. If w tak two prmutatios as grators, th w ca at most obtai th symmtric group S. Lmma 3 (Two prmutatios). Lt 2. If α, β S, th α, β < V. Sktch of Proof. Obviously, for prmutatios α ad β w hav α, β!. I ordr to prov th statd iquality it suffics to show that! < V 1. Th dtails ar lft to th radr. Nxt w cosidr th cas of two o-bijctiv trasformatios. Lmma 4 (Two o-bijctiv trasformatios). Lt 2. If both α ad β i T \ S, th α, β < V. Proof. Sic α ad β ar both o-bijctiv, thr ar idics j 1 < k 1 ad j 2 < k 2 such that (j 1 )α = (k 1 )α ad (j 2 )β = (k 2 )β. I this cas w ca costruct a prmutatio π such that (i 1 )π = j 1, (i 1 + 1)π = k 1 for som idx i 1 ad (i 2 )π = j 2, (i 2 + 1)π = k 2 for som idx i 2. If j 1 = j 2, th it is th cas that i 1 = i 2, similarly if j 1 = k 2, th i 1 = i 2 + 1, tc. This mas that all trasformatios gratd by παπ 1 ad πβπ 1 satisfy th scod part of Dfiitio 2 for d = 1. Accordig to Dfiitio 2 th st παπ 1, πβπ 1, ad thrfor also α, β which is isomorphic, hav lss lmts tha V 1, sic at last th prmutatios ar missig. Thus, th statd claim follows. 5 Smigroup Siz Two ad Mor Cycls Fially w cosidr th cas whr o of th grators is a prmutatio α cosistig of two or mor cycls ad th othr is a o-bijctiv trasformatio. I this cas w distiguish two sub-cass, accordig to whthr th o-bijctiv trasformatio β mrgs lmts from th sam or diffrt cycls of α. W start our ivstigatio with th cas whr thr ar i ad j such that (i)β = (j)β ad both ar locatd withi th sam cycl of α.
Lmma 5 (A arbitrary prmutatio ad a o-bijctiv mappig mrgig lmts from th sam cycl). Thr xists a atural umbr N such that for vry N th followig holds: Lt α, β T b trasformatios whr α is a prmutatio. Furthrmor lt β b a o-bijctiv trasformatio such that (i)β = (j)β ad both i ad j ar locatd i th sam cycl of α. Th thr xist k ad l with = k + l such that α, β < U k,l. Proof. W assum that i ad j ar locatd i th sam cycl of lgth m with distac d w.r.t. thir locatio withi th cycl. W ca assum that d divids m, othrwis w ca fid a isomorphic smigroup whr this is th cas, followig th idas of th proof of Lmma 1. With a similar argumt as i th proof of Thorm 3 w ca dduc that th smigroup gratd by α ad β cotais at most som prmutatios ad th ivalid colourigs of a graph G, whr G cosists of d circls of lgth m d ad m isolatd ods. Th umbr of valid colourigs of such a graph quals (( 1) m d + ( 1) m d ( 1)) d m. Thrfor w coclud α, β +! ( ( 1) m d + ( 1) m d ( 1) ) d m. Similar rasoig as i th proof of Thorm 4 shows that +! ( ( 1) m d + ( 1) m d ( 1) ) d m +! ( ( 1)( 2) 2 ) 3 ad lim ( ) +! ( 1)( 2) 3 2 = 1 1. Hc, a similar asymptotic argumt as i th proof of Thorm 5 shows that thr is a atural umbr N such for vry N th siz of th smigroups o lmts udr cosidratio is strictly lss tha th siz of U k,l, for suitabl k ad l with = k + l. Fially, w cosidr th cas whr th o-bijctiv trasformatio β mrgs lmts from diffrt cycls of th prmutatio α. I th rmaidr of this sctio w assum = k +l to b a prim umbr. Th rasos for this assumptio is that k ad l ar always coprim, which guarats that U k,l ca b gratd by two grators oly. Lmma 6 (A prmutatio with two or mor cycls ad a o-bijctiv mappig mrgig lmts from diffrt cycls). Lt b a prim umbr ad lt α, β T b trasformatios whr α is a prmutatio cosistig of m 2 cycls. Furthrmor lt β b a o-bijctiv trasformatio such that (i)β = (j)β ad i ad j ar locatd i diffrt cycls of α. Th thr xist k ad l with = k + l such that α, β U k,l.
Proof. W dfi U := α, β ad show that U U, whr U is gratd by a two-cycl prmutatio α ad a o-bijctiv mappig β that mrgs lmts of diffrt cycls, as dscribd blow i dtail. Now assum that th m cycls i α hav lgths k 1,..., k m, i.., = m i=1 k i. Furthrmor th sts of lmts of th m cycls ar dots by C 1,..., C m ad C i = k i. Without loss of grality w may assum that β mrgs lmts of th first two cycls C 1 ad C 2. W ow cosidr th followig two cass accordig to which lmt is missig i th imag of β: 1. Thr is a lmt h which is ot cotaid i th imag of β ad morovr, h is ot locatd i th first two cycls of α. So lt us assum that it is locatd i th third cycl C 3. Lt α b a prmutatio with two cycls, whr th lmts of th first cycl ar C 1 = C 2 m i=4 C i ad th lmts of th scod cycl ar C 2 = C 1 C 3. I th cycls ths lmts ca b arragd i a arbitrary way. W ow st k = k 2 + m i=4 k i ad l = k 1 + k 3. Sic = k + l ad is prim, it follows that gcd{k, l} = 1. Similar to th costructio for th U k,l o ca ow fid a trasformatio β such that α ad β grat a smigroup U isomorphic to U k,l. That mas, th lmts of U ar xactly th multipls of α ad all trasformatios γ which satisfy (i)γ = (j)γ, for i C 1 ad j C 2, ad whr at last o lmt of C 2 is missig i th imag of γ. Now lt us compar th sizs of U ad U. First cosidr oly th obijctiv trasformatios of U. This icluds at last all o-bijctiv trasformatios gratd by α ad β, sic th first cycl of α icluds C 2 ad th scod cycl of α icluds C 1 ad C 3. So for ay o-bijctiv γ gratd by α ad β thr ar idics i C 1, j C 2, h C 3 such that (i)γ = (j)γ ad h img(γ). This implis that γ ca b gratd by α ad β as wll. Howvr, U may cotai mor prmutatios tha U. I th worst cas, if gcd{k i, k j } = 1 for all pairs of cycl lgths with i j, th U cotais m i=1 k i prmutatios, whras U cotais oly kl prmutatios, which might b lss. W show that this shortcomig is alrady compsatd by th umbr of trasformatios with imag siz 1. Th smigroup U cotais k 1 k 2 k 3 ( 1)! mappigs with imag siz 1. W first choos th two lmts which ar i th sam krl quivalc class, for which thr ar k 1 k 2 possibilitis, th w choos th lmt of th imag that is missig, for which thr ar k 3 possibilitis, ad fially w distribut th 1 lmts of th imag oto th krl quivalc classs. I th sam way w ca show that thr ar kl 2 ( 1)! trasformatios with imag siz 1 i U. Now dfi k = m i=4 k i ad obsrv, that k might b qual to 0. Th w coclud that kl 2 k 1 k 2 k 3 = (k 2 + k )(k 1 + k 3 ) 2 k 1 k 2 k 3 = (k 2 + k )(k 2 1 + 2k 1 k 3 + k 2 3) k 1 k 2 k 3 = k 2 1k 2 + k 1 k 2 k 3 + k 2 k 2 3 + k k 2 1 + 2k k 1 k 3 + k k 2 3 k 1 + k 2 + k 3 + k =.
Thrfor U cotais at last! mor trasformatios of imag siz 1 tha U. This maks up for th missig prmutatios, sic thr ar at most! of thm. 2. Th missig lmt h of th imag of β is locatd i o of th first two cycls. Th a aalogous costructio as i (1) shows how to costruct suitabl α ad β such that U α, β. Du to th lack of spac th dtails ar lft to th radr. This complts our proof ad shows that α, β U k,l, bcaus i both cass smigroup U is isomorphic to som U k,l, for appropriat k ad l. 6 O th Maximality of U k,l Smigroups Now w ar rady to prov th mai thorm of this papr, amly that th siz maximal smigroup has U k,l lmts, for som k ad l, whvr = k + l is a prim gratr or qual tha 7. Obsrv, that th followig thorm strgths Lmma 5. Thorm 6. Lt 7 b a prim umbr. Th th smigroup U k,l, for som k ad l with = k + l, is maximal w.r.t. its siz amog all smigroups which ca b gratd with two grators. Proof. Sic all othr cass hav alrady b tratd i th Lmmata 3, 4, ad 6, it is lft to show that U k,l has mor lmts tha th smigroup V, whr V is gratd by α ad β ad lattr mappig mrgs lmts locatd i th sam cycl of α. Not that k ad l ar trivially coprim whvr = k + l is a prim. W hav show i Lmma 5 that V +! ( ( 1)( 2) 3 Furthrmor from [3] it follows that U k,l ) 3 = +! (( 1)( 2)) 3. ( ) l! k l ( ) k k l l. l W us Stirlig s approximatio i th vrsio ( ) ( ) 1 2π <! < 2π 12 giv i [1, 11]. I this way w obtai a uppr boud for V ad a lowr boud for U k,l, s th proof i [3], as follows: ad V + ( ) 1 2π 12 (( 1)( 2)) 3
U k,l ( ( ) ) 2 2 2 1 1 12 + 8 1 12. Th uppr boud for V is smallr tha th lowr boud for U k,l whvr 2 ( 2 ) 2 1 12 + 8 1 1 12 < ( ( 1)( 2) 2 ) 3 } {{ } A() ( ) 1 2π 1 12 }{{} B() Th fuctio A() is mooto ad covrgs to 1 0.3678794412 whil th fuctio B() is atito ad covrgs to 0. For 20 w hav A() > 0.358 ad B() < 10 7, ad thrfor A() B() > 0.35 =: c. W st c 1 = 0.01 ad c 2 = 0.34 ad solv th quatios 2 ( 2 ) 2 1 12 < c1 ad 8 1 1 12 < c2. Ths quatios ar satisfid if ( ) log c 1 2 1 1 12 > 2 log 2 32.81753852 ad > ( 8 c 2 1 12 ) 2 81.75504594, i.., whvr 82. Th rmaiig cass for 7 81 hav b chckd with th hlp of th Groups, Algorithms ad Programmig (GAP) systm for computatioal discrt algbra. To this d w hav vrifid that V U k,l, for som k ad l, whr th uppr boud for V from Lmma 5 ad th xact valu of U k,l was usd. 2 It turd out that V is maximal w.r.t. siz for all V smigroups.. 7 Coclusios W hav cofirmd th cojctur i [3] o th siz of two grator smigroups. I th d, w hav show that for prim, such that 7, th smigroup gratd by two grators with maximal siz ca b charactrizd i a vry ic ad accurat way. Th cass 2 6 ar ot tratd i this papr, but w 2 Th formula giv blow did ot appar i [3] ad givs th xact siz of th U k,l smigroup: Lt k, l N such that gcd{k, l} = 1. Th smigroup U k,l cotais xactly U k,l = kl + X i=1! i!! ( ) l i l i ( )( )! ix k l i! r i r lmts, whr = k + l. Hr i stads for th Stirlig umbrs of th scod kid ad dots th umbr of possibilitis to partitio a -lmt st ito i o-mpty substs. r=1
wr abl to show that i all ths cass th smigroup V cotais a maximal umbr of lmts. Hr 2 5 wr do by brut forc sarch usig th GAP systm ad = 6 by additioal quit ivolvd cosidratios, which w hav to omit to du th lack of spac. Morovr, w hav compltly classifid th cas wh o grator is a prmutatio cosistig of a sigl cycl. Nvrthlss, som qustios rmai uaswrd. First of all, what about th cas wh 7 is ot a prim umbr. W cojctur, that Thorm 6 also holds i this cas, but w hav o proof yt. Also, th qustio how to choos k ad l proprly rmais uaswrd. I ordr to maximiz th siz of U k,l o has to miimiz th umbr of valid colourigs s [3] which is miimal if k ad l ar clos to 2. This clashs with th obsrvatio that th cycl α from which a lmt i th imag of β is missig should b as larg as possibl. Nvrthlss, to maximiz th siz of U k,l w cojctur that for larg ough both k ad l ar as clos to 2 as th coditio that k ad l should b coprim allows. Agai a proof of this statmt is still missig. I ordr to udrstad th vry atur of th qustio much bttr, a stp towards its solutio would b to show that th squc U k,l for fixd = k+l ad varyig k is uimodal. 8 Ackowldgmts Thaks to Rob Johso, ad Paul Pollack, ad Joh Robrtso for thir hlp i maagig th sstial stp i th proof of Lmma 1. Rfrcs 1. W. Fllr. Stirlig s formula. I A Itroductio to Probability Thory ad Its Applicatios, volum 1, chaptr 2.9, pags 50 53. Wily, 3rd ditio, 1968. 2. G. H. Hardy ad E. M. Wright. A Itroductio to th Thory of Numbrs. Clardo, 5th ditio, 1979. 3. M. Holzr ad B. Köig. O dtrmiistic fiit automata ad sytactic mooid siz. I M. Ito ad M. Toyama, ditors, Prprocdigs of th 6th Itratioal Cofrc o Dvlopmts i Laguag Thory, pags 229 240, Kyoto, Japa, Sptmbr 2002. Kyoto Sagyo Uivrsity. To appar i LNCS. 4. J. M. Howi. A Itroductio to Smigroup Thory, volum 7 of L. M. S. Moographs. Acadmic Prss, 1976. 5. E. Ladau. Übr di Maximalordug dr Prmutatio ggb Grads. Archiv dr Mathmatik ud Physik, 3:92 103, 1903. 6. J.-L. Nicolas. Sur l ordr maximum d u élémt das l group s ds prmutatios. Acta Arithmtica, 14:315 332, 1968. 7. J.-L. Nicolas. Ordr maximum d u élémt du group d prmutatios t highly composit umbrs. Bullti of th Mathmatical Socity Frac, 97:129 191, 1969. 8. S. Piccard. Sur ls bass du group symétriqu t ls coupls d substitutios qui gdrt u group régulir. Librairi Vuibrt, Paris, 1946. 9. J.-E. Pi. Varitis of formal laguags. North Oxford, 1986. 10. R. C. Rad. A itroductio to chromatic polyomials. Joural of Combiatorial Thory, 4:52 71, 1968. 11. H. Robbis. A rmark of Stirlig s formula. Amrica Mathmatical Mothly, 62:26 29, 1955.