LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Polya Theory by MapleKahtan H. Alzubaidy Key Words : Permutation Groups, Burnside's Theorem, Polya's Theorems Introduction In this application of Maple a few procedures are written to do the computations related to Burnside's counting theorem and also to Polya's theorems. Everything is done in terms of permutation groups. Permutations Group A permutation for example 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 (1,2)(3,4) is represented by disjoint cycles 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 The identity of degree 4 , for example is given by [1,2,3,4] or [[1],[2],[3],[4]]. A permutation group is denoted by a list of its permutations.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EocmVzdGFydEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYvUSI7RidGL0Y1RjgvJSZmZW5jZUdGNC8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==QyYtSSV3aXRoRzYiNiNJKWNvbWJpbmF0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiLUYkNiNJKkxpc3RUb29sc0dGKEYr A product of two permutations S and T of degree n.p:=proc(S,T,n)[seq(S[T[r]],r=1..n)];end proc;JSFHJSFH Basic Examples of Permutation Groups Trivial group of degree 4 Id4:=[[1,2,3,4]]; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRitGK0YrLUkjbWlHRiQ2J1EkQWxsRidGLy8lJ2l0YWxpY0dGOUYyRjVGKy1GTDYnUS1wZXJtdXRhdGlvbnNGJ0YvRk9GMkY1RitGL0YyRjU=(Symmetric Group S(n) of degree n )LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=S:=proc(n)permute(n);
end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyQtSSJTRzYiNiMiIiJGJw==QyQtSSJTRzYiNiMiIiMiIiI=QyQtSSJTRzYiNiMiIiQiIiI=QyQtSSJTRzYiNiMiIiUiIiI=QyQ+SSNaMkc2IjckNyQiIiIiIiM3JEYpRihGKA==QyQ+SSNaM0c2IjclNyUiIiIiIiMiIiQ3JUYpRipGKDclRipGKEYpRig=QyQ+SSNaNEc2IjcmNyYiIiIiIiMiIiQiIiU3JkYpRipGK0YoNyZGKkYrRihGKTcmRitGKEYpRipGKA==QyQ+SSJWRzYiNyY3JiIiIiIiIyIiJCIiJTcmRilGKEYqRis3JkYoRilGK0YqNyZGKUYoRitGKkYoQyQ+SSNENEc2IjcqNyYiIiIiIiMiIiQiIiU3JkYpRipGK0YoNyZGKkYrRihGKTcmRitGKEYpRio3JkYpRihGK0YqNyZGK0YqRilGKDcmRipGKUYoRis3JkYoRitGKkYpRig= The above examples can be put in terms of disjoint cycles including 1-cycles as followsJSFHQyQ+SSVDSWQyRzYiNyM3JDcjIiIiNyMiIiNGKQ==QyQ+SSVDSWQzRzYiNyM3JTcjIiIiNyMiIiM3IyIiJEYpLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=QyQ+SSVDSWQ0RzYiNyM3JjcjIiIiNyMiIiM3IyIiJDcjIiIlRik=QyQ+LUkjQ1NHNiI2IyIiIzckNyQ3IyIiIjcjRig3IzckRihGLEYsQyQ+LUkjQ1NHNiI2IyIiJDcoNyU3IyIiIjcjIiIjNyNGKDckRis3JEYoRi43JDckRi5GLEYvNyM3JUYuRihGLDcjNyVGKEYsRi43JDckRihGLEYtRiw=QyQ+SSRDWjJHNiI3JDckNyMiIiI3IyIiIzcjNyRGK0YpRik=QyQ+SSRDWjNHNiI3JTclNyMiIiI3IyIiIzcjIiIkNyM3JUYrRi1GKTcjNyVGLUYpRitGKQ==QyQ+SSRDWjRHNiI3JjcmNyMiIiI3IyIiIzcjIiIkNyMiIiU3IzcmRitGLUYvRik3JDckRilGLTckRitGLzcjNyZGL0YpRitGLUYpQyQ+SSNDVkc2IjcmNyY3IyIiIjcjIiIjNyMiIiQ3IyIiJTclRixGLjckRitGKTclRihGKjckRi9GLTcjNyZGK0YpRi9GLUYpQyQ+SSRDRDRHNiI3KjcmNyMiIiI3IyIiIzcjIiIkNyMiIiU3IzcmRilGK0YtRi83JDckRilGLTckRitGLzcjNyZGL0YpRitGLTckNyRGKUYrNyRGLUYvNyQ3JEYpRi83JEYrRi03JUYoRixGNDclRjNGKkYuRik=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Burnside Counting Theorem Fixed Points of a PermutationFixedPoints:=proc(L){select(x->is (x=L[x]),L)[]};end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Group Action Action of a permutation L of degree n on element x in {1,2,...,n}.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=action:=proc(L,x)L[x];end proc; Orbit of x in a permutation group GLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=orbit:=proc(x,G){seq(action(L,x),L in G)};end proc;QyQtSSZvcmJpdEc2IjYkIiIjLUkiU0dGJTYjIiIkIiIiJSFH Burnside counting theorem Number of orbits = 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 G is a permutation group of degree nBurnside:=proc(G,n)1/nops(G)*add(nops(FixedPoints(g)),g = G);end proc;QyQtSSlCdXJuc2lkZUc2IjYkNyM3JiIiIiIiIyIiJCIiJUYsRik=QyQtSSlCdXJuc2lkZUc2IjYkNyQ3JiIiIiIiIyIiJCIiJTcmRipGKUYrRixGLEYpQyQtSSlCdXJuc2lkZUc2IjYkLUkiU0dGJTYjIiIkRioiIiI=QyQtSSlCdXJuc2lkZUc2IjYkSSJWR0YlIiIlIiIiQyQtSSlCdXJuc2lkZUc2IjYkSSNENEdGJSIiJSIiIg==QyQtSSlCdXJuc2lkZUc2IjYkSSNaM0dGJSIiJCIiIg== Polya Theorems Cycle Index Polynomial ZG of a permutation group G of degree n. ZG(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where 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 In the following procedurs permutation groups should be given in cycle notationsLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=expo:=proc(L)[seq(nops(op(i,L)),i=1..nops(L))];end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=l:=proc(L,m,t) nops(select(x-> is(nops(x)=t),L));end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=exponents:=proc(L,m) [seq(l(L,m,t),t=1..m)];end proc;JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=vars:=proc(m) [seq(x[i],i=1..m)];end proc; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=mono:=proc(L,m) local B,E; B:= [seq(x[i],i=1..m)]; E:=exponents(L,m); product((x[r])^E[r],r=1..m);end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ZG:=proc(G,m) 1/nops(G) *add(mono(L,m),L=G);end proc;QyQtSSNaR0c2IjYkSSVDSWQyR0YlIiIjIiIiQyQtSSNaR0c2IjYkSSVDSWQzR0YlIiIkIiIiQyQtSSNaR0c2IjYkSSVDSWQ0R0YlIiIlIiIiQyQtSSNaR0c2IjYkLUkjQ1NHRiU2IyIiI0YqIiIiQyQtSSNaR0c2IjYkLUkjQ1NHRiU2IyIiJEYqIiIiQyQtSSNaR0c2IjYkSSRDWjNHRiUiIiQiIiI=QyQtSSNaR0c2IjYkSSNDVkdGJSIiJSIiIg==QyQtSSNaR0c2IjYkNyQ3JTcjIiIiNyMiIiM3IyIiJDckRik3JEYuRixGLkYqLUkjWkdHNiI2JEkkQ0Q0R0YkIiIl Let D and R be two finite sets. 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 Enumeration Theorem The number of equivalence classes induced by 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:=proc(G,d,r) ZG(G,d); subs([seq(x[i] = r, i = 1 .. d)],ZG(G,d));end proc;QyQtSSdQb2x5YTFHNiI2JUklQ0lkNEdGJSIiJUkickdGJSIiIg==QyQtSSdQb2x5YTFHNiI2JS1JI0NTR0YlNiMiIiNGKkkickdGJSIiIg==QyQtSSdQb2x5YTFHNiI2JS1JI0NTR0YlNiMiIiRGKkkickdGJSIiIg==QyQtSSdQb2x5YTFHNiI2JUkkQ1ozR0YlIiIkSSJyR0YlIiIiQyQtSSdQb2x5YTFHNiI2JUkjQ1ZHRiUiIiVJInJHRiUiIiI=QyQtSSdQb2x5YTFHNiI2JUkkQ0Q0R0YlIiIlSSJyR0YlIiIi Inventory Polynomials Let D and R be two finite sets. The weight of r\342\210\210R is a symbol w(r) associated with r. The weight of a function 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 is defined by w(f)=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 that w(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 The inventory of F\342\212\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 Theorem. Let 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 Polya Fundamental Theorem. Let D and R be two finite sets and G a permutation group of D. Then Inv(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 The following procedure computes the inventory polynomial of a permutation group G of degree d in terms of the symbols 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Polya2:=proc(G,d,r) seq(a[i],i=1..r); ZG(G,d); subs([seq(x[t] = sum(a[i]^t, i = 1 .. r),t=1..d)],ZG(G,d));end proc;QyQtSSdQb2x5YTJHNiI2JUklQ0lkNEdGJSIiJSIiJCIiIg==QyQtSSdQb2x5YTJHNiI2JS1JI0NTR0YlNiMiIiNGKkYqIiIiJSFHQyQtSSdQb2x5YTJHNiI2JS1JI0NTR0YlNiMiIiNGKiIiJCIiIg==QyQtSSdleHBhbmRHJSpwcm90ZWN0ZWRHNiNJIiVHNiIiIiI=QyQtSSdQb2x5YTJHNiI2JS1JI0NTR0YlNiMiIiRGKkYqIiIiQyQtSSdleHBhbmRHJSpwcm90ZWN0ZWRHNiNJIiVHNiIiIiI=QyQtSSdQb2x5YTJHNiI2JkkjQ1ZHRiUiIiUiIiNGKSIiIg==QyQtSSdQb2x5YTJHNiI2JUkkQ0Q0R0YlIiIlIiIkIiIiJSFH Let D and R be two finite sets. Suppose that G is a permutation group of degree 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 Polya Generalized Theorem(DeBruijn) The number of equivalence classes induced by 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 The following procedures lead to the computation of the number of these equivalence classes.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= The powers LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUklbXN1cEdGJDYlLUkjbWlHRiQ2J1EiaEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2KC1GLzYnUSJyRidGMkY1RjhGO0YyLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGNEY4RjsvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjJGOEY7pr:=proc(h,p,n,r)local q;q:=t->p(q(t-1),h,n);q(1):=h;q(r);end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= List of numbers of fixed points of h, 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R:=proc(h,p,n,m)[seq(nops(FixedPoints(pr(h,p,n,r))),r=1..m)]end proc;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= List of numbers of fixed points of 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:=proc(H,n,m)[seq(R(h,p,n,m),h in H)];end proc;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pgg:=proc()local m,PP;m:=nargs-1;PP:=PG(args[1],m);subs([seq(x[i-1]=args[i],i=2..m+1)],PP);end proc;JSFHJSFH G of degree m in cycles and H of degree n in permutation listsPolya3:=proc(G,m,H,n)local RRR;RRR:=RR(H,n,m);1/nops(H) *add(z,z in [seq(pgg(G,RRR[i][]),i=1..nops(H))]);end proc;QyQtSSdQb2x5YTNHNiI2Ji1JI0NTR0YlNiMiIiRGKi1JIlNHRiU2IyIiI0YuIiIiQyQtSSdQb2x5YTNHNiI2Ji1JI0NTR0YlNiMiIiRGKi1JIlNHRiVGKUYqIiIiQyQtSSdQb2x5YTNHNiI2Ji1JI0NTR0YlNiMiIiRGKjcjNyQiIiIiIiNGLkYtQyQtSSdQb2x5YTNHNiI2JkkkQ0Q0R0YlIiIlNyM3JiIiIiIiIyIiJEYoRihGKw==QyQtSSdQb2x5YTNHNiI2Ji1JI0NTR0YlNiMiIiQiIiU3IzcmIiIiIiIjRipGK0YrRi4=QyQtSSdQb2x5YTNHNiI2Ji1JI0NTR0YlNiMiIiRGKjcjNyUiIiIiIiNGKkYqRi0=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Generalized Polya's Fundamental Theorem(DeBruijn) N = 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 that D ={1,2,..., m} and R ={ 1,2,...,n}.One may program this formula by Maple to complete the procedures. Department of Mathematics, Faculty of Science, University of Benghazi, Libya E-mail: kahtanalzubaidy@yahoo.comJSFHJSFH