Matrix representation of a solution of a combinatorial problem of the group theory

Transcription

1 Matrix represetatio of a solutio of a combiatorial problem of the group theory Krasimir Yordzhev, Lilyaa Totia Faculty of Mathematics ad Natural Scieces South-West Uiversity 66 Iva Mihailov Str, 2700 Blagoevgrad, Bulgaria yordzhev@swubg, liliaa_totia@abvbg Abstract: A equivalece relatio i the symmetric group, where ³ 2 is a positive iteger has bee cosidered A algorithm for calculatio of the umber of the equivalece classes by this relatio for arbitrary iteger has bee described Keywords: Symmetric Group, Equivalece Relatio, Equivalece Classes, Directed Graph, Euler Fuctio, C++ Programmig Laguage S 1 INTRODUCTION The object of the preset study is to be received a algorithm for computer calculatio of some combiatorial characteristics of the symmetric group This algorithm has bee based o the theoretical elaboratios described closely i [3] ad [4] I our study with we deote the set of the positive itegers Let Î The with we deote the set = {1, 2,, With ì 1 2 ü S ï æ ö = is, s 1,2,,, is it fors t ï í Î = ¹ ý i1 i2 i ¹ ï çè î ø ïþ we deote the symmetric group o the set, i e the group of all oe-tooe mappigs a : of the set i oeself If mî, the with GCD(, m) we deote the greatest commo divisor of itegers ad m The ad m are relatively primes if ad oly if whe GCD(, m) =1

2 With j( ) we deote the Euler fuctio, i e the umber of elemets of that are relatively primes with (see more for example i [1] or [5]) By defiitio j(1) = 1 2 PRIOR INFORMATION Let be a positive iteger We cosider a elemet æ1 2 1 ö s - = ÎS ç çè2 3 1 ø I [3] the followig equivalece relatio had bee itroduced i : we say that abî, S are s - equivalet if the itegers ad l exist such that l a= s bs, that is equivalet to coditio that itegers ad l 1 1 exist such that s 1 l a = bs 1 The followig tas has bee put: To fid the umber of the equivalece classes by so defied equivalece relatio, with aother words the cardiality of the factor set Q = S/ s To be solved so puttig tas a directed graph with the set of vertices G {,, V Ì = a b a bî, 2 has bee costructed, formed by the ext mea: 1 For every iteger m Î such that 1, m V 2 If l, Î V, the divides GSD(, m ) = 1 Î exists, as i this vertex arcs do ot eter S a vertex 3 Let l, Î V ad let p is a prime divisor of The we receive the umber r that is equal to the remaider after the multiplicatio pl is divided ito We costruct a vertex p r Î V ad a arc with begi the vertex l, ad ed the, vertex p, r 4 Aother vertex ad aother arc i the graph G acceptig received by the mea described above do ot exist For example at = 12 the graph G12 is show o figure 1:

3 Fig 1: The graph G 12 We itroduce the followig partial order i the vertices set of the graph G : If v1, v2 ÎV, the v1< v2 if ad oly if whe a directed path with begi vertex v 1 ad ed vertex v 2 exists It is easily to see that by so itroduced order is semilattice with the uique maximal elemet Î V V V, ad j( ) i umber miimal elemets, each of id 1, m Î V, where j () is the Euler fuctio For more details of the graph theory see for example i [6] For each vertex l, Î V we defie the fuctio h (, ) that depeds o ad, but does ot deped o l 1 1 æ - æö ö h (, ) = ç 1! rhr (, ) - - ç å ç è è ø rt, < l, ø (1) ç( ) The followig assertio had bee demostrated i [3]: Theorem 1 [3] The umber of the equivalece classes by the s - equivalece is equal to (2) Q = å h (, ) l, ÎV Theorem 1 gives us a effective algorithm for the maual calculatio of Q Costructio of the graph G is ecessary for this object This approach gives relatively good results at relatively small values of as the experiece of authors has bee show With icrease of

4 probability of errors icreases repeatedly, because of the umber of classes icreases expoetially, accordig to formulas (1) ad (2) It had bee demostrate i [4] that all assertios cosidered i [3] are valid for a arbitrary elemet s Î S, o coditio that s is a cycle with legth It is easy to see that we ca reorgaize the formula (1) i the followig recursive id: (3) ç( ) æ ö -1 1 ç æö h (, ) = -1! - rt( r,, ) hr (, ) ç å for > 1 ç è ø r çè r< ø ad (4) h (,1) = 1, where the fuctio t( r,, ) gives as the umber of vertices rs, Î V, such that rs, <, Î V The followig assertio had bee demostrated i [4]: Theorem 2 [4] The umber of the vertices of id l, Î V is equal to j, where j m is the Euler fuctio ( ) ( ) As a cosequece of theorem 2 the formula (2) has bee reorgaized i the followig id: (5) Q = å h (, ) jç æ ö çè ø 3 MATRIX REPRESENTATION AND PROGRAM REALIZATION I this sectio we use program laguage C++ for descriptio of algorithms cosidered i the preset study The program was bee tested i a programmig eviromet Borlad C++ Builder

5 is a global costat iteger pa- ad may be used without explicit At the examples we cosider that rameter, equal to the order of the group S declaratio For usig theorem 1 as we avail ourselves of formulas (3) ad (4) it is coveiet to fill the elemets of the matrix M = m, cosistig of 4 rows ( ij ad q colums, where q is equal to the umber of all positive divisors of the iteger (icludig 1 ad ) Never the less the iteger will be writte dow i M, i our program realizatio we declare M as two-dimesioal array of type double because of fact that we expect to receive too large values at calculatio of the fuctio h (, ) accordig to formula (3) I the first row we write dow all itegers dividig without remaider the parameter That ca be realized with the help of the followig procedure, for example: void Divisors(it, it FirstRow[], it& q) { q = 1; FirstRow[0] = 1; for (it t = 2 ; t <= ; t++) { if (%t == 0) { FirstRow[q] = t; q++; As additioal effect of the wor of the fuctio Divisors we receive the umber q of all divisors of the global parameter It the ed the procedure receives a iteger array FirstRow, correctly filled with the divisors of We copy the values of the array FirstRow to first row of the array M We fill the secod row of M with the values of the Euler fuctio j ( ), where has bee tae from the correspodig compoet of the first row of M The fuctio j( m) ca be realized as we use a variety of the algorithm, ow as the Sieve of Eratosthees [2,5] it EulerFuctio(it m) { it t =1; it b[100]; for (it i=1; i <= m; i++) { b[i] = i; )

6 for (it i=2; i <= m; i++) { if (b[i] == 0) { cotiue; if (m%b[i] == 0) { for (it j=1; i*j<=m; j++) { b[i*j] = 0; else t++; retur t; We fill the third row of M with the values of the fuctio h) (,, accordig to formulas (3) ad (4), where has bee tae from the correspodig compoet of the first row of M Here we cosider agai that is a global costat parameter, ie it is ot ecessary to be preset i the list of formal parameters of the fuctio described below For simplicity of the expositio we will ot verify correctess of the data, ie whether divides This coditio has bee esured by the fact that has bee tae from the first row of the matrix M, where o coditio oly divisors of has bee writte dow We declare the type of the fuctio h(it), as well as some worig variables, which values are itegers with the type log double because of expectatio to receive to large values accordig to formula (3) For the calculatio of the fuctio h (, ), must first be calculated the fuctio t( r,, ) This ca be doe usig the followig algorithm: // Euclidea algorithm for fidig the greatest commo divisor of itegers a ad b: it GCD(it a, it b) { while (a!= b) if (a > b) { a = a-b; else b = b-a; retur a; //

7 // Fuctio examied whether l, is the vertex of V bool Vertex(it, it l, it ) { if ((<=0) (>) (l>) (%!= 0)) { retur false; else { if (( == )) { if (l == ) retur true; else for (it m=1; m<=; m++) { if ((GCD(m,) == 1) && (m*% == l)) retur true; retur false; // // The fuctio t( r,, ): it tau(it, it, it r) { it t=0; it p = /r; for (it s=1; s<; s++) { if (Vertex(r,s,) && (s*p% == %)) { t++; retur t; The a algorithm for obtaiig h (, ) accordig to formulas (3) ad (4) ca be realized usig the followig C++ fuctio: log double h(it, it ) { log double t; if ( == 1) { t=1;

8 else { log double fact = 1; for (it i=2; i <= -1; i++) { fact = fact*i; log double pow = 1; it p = /; for (it i=1; i<=-1; i++) { pow = pow*p ; log double sum = 0; for (it r=1; r<=-1; r++) { if (%r == 0) { sum = sum+r*tau(,,r)*h(,r); t = (fact*pow-sum)/; retur t; We receive the fourth row of M as multiply compoet by compoet the secod with the third rom Here we will sip the descriptio of the mai fuctio ad the iput ad output operatios as they are specific to differet programmig eviromets At = 12 the filled matrix M is show i table 1: Tab 1: Matrix M at = j ( ) h (, ) m j m3 j W e receive the fial result for the umber of the equivalece classes of S by the cosidered equivalece relatio as we add the elemets of the last row of the matrix M accordig to formula (5) We calculate from table 1: Q 12 = = Q

9 We show i table 2 the values of Q at 19 calculated by the algorithm described above: Tab 1: The umber of the 19 equivalece classes i cocerig to equivalece at Q Q Q S REFERENCES [1] Adreescu, T, Adrica, D, Feg, Z (2007) 104 Number Theory Problems Bosto: Birhauser [2] Reigold E W (1977) Combiatorial Algorithms Theory ad Practice New Jersey, Pretice-Hall [3] Yordzhev, K (2004) O a equivalece relatio i the set of the permutatio matrices, i Discrete Mathematics ad Applicatios, Blagoevgrad, SWU N Rilsi, [4] Yordzhev, K (to appear) O a Combiatorial Problem i the Symmetric Group [5] Мирчев, И, (1995) Теория на числата, Благоевград, ЮЗУ Н Рилски [6] Мирчев И, (2001) Графи Оптимизационни алгоритми в мрежи, Благоевград, ЮЗУ Н Рилски