> #ok to post ; > #Yifan Zhang, 10/10/2022 ; > ; > ; > #Q1. ; > # ; > # ; > # ; > with(combinat) [Chi, bell, binomial, cartprod, character, choose, composition, conjpart, decodepart, encodepart, eulerian1, eulerian2, fibonacci, firstcomb, firstpart, firstperm, graycode, inttovec, lastcomb, lastpart, lastperm, multinomial, nextcomb, nextpart, nextperm, numbcomb, numbcomp, numbpart, numbperm, partition , permute, powerset, prevcomb, prevpart, prevperm, randcomb, randpart, randperm , rankcomb, rankperm, setpartition, stirling1, stirling2, subsets, unrankcomb, unrankperm, vectoint] ; > P1:=randperm(9); P2:=randperm(9); Typesetting:-mprintslash([(P1 := [3, 8, 4, 9, 1, 5, 6, 7, 2])],[[3, 8, 4, 9, 1, 5, 6, 7, 2]]) Typesetting:-mprintslash([(P2 := [8, 5, 6, 4, 3, 9, 7, 2, 1])],[[8, 5, 6, 4, 3, 9, 7, 2, 1]]) ; > #P1*P2 = [3,8,9,5,1,6,7,4,2] ; > #P2*P1 = [5,4,8,3,2,1,7,6,9] ; > ; > MulPers:=proc(P1,P2) local n,i: > #The first few lines checks that the input is correct > if not (type(P1,list) and type(P2,list)) then > print(`Bad input`): > RETURN(FAIL): > fi: > n:=nops(P1): > > if not (convert(P1,set)={seq(i,i=1..n)} and convert(P2,set)={seq(i,i=1..n)} ) then > print(`Bad input`): > RETURN(FAIL): > fi: > > [seq(P2[P1[i]],i=1..n)]: > > > end: > MulPers(P1,P2); [6, 2, 4, 1, 8, 3, 9, 7, 5] ; > MulPers(P2,P1); [7, 1, 5, 9, 4, 2, 6, 8, 3] ; > ; > P3:= randperm(9); Typesetting:-mprintslash([(P3 := [2, 4, 1, 5, 8, 9, 7, 3, 6])],[[2, 4, 1, 5, 8, 9, 7, 3, 6]]) ; > #P^-1 = [2,5,3,9,4,8,1,6,7]; > InvPer:=proc(P) local i,n,T: > if not type(P,list) then > print(`Bad input`): > RETURN(FAIL): > fi: > > n:=nops(P): > if not convert(P,set)={seq(i,i=1..n)} then > print(`Bad input`): > RETURN(FAIL): > fi: > > #Recalling the list P is the function that maps i to P[i], we define a TABLE (a Maple Data structure) that > #maps P[i] to i > > for i from 1 to n do > T[P[i]]:=i: > od: > > #We convert it back into a list > [seq(T[i],i=1..n)]: > end: > InvPer(P3) [3, 1, 8, 2, 4, 9, 7, 5, 6] ; > ; > P4:=randperm(9); Typesetting:-mprintslash([(P4 := [4, 6, 9, 1, 3, 7, 5, 8, 2])],[[4, 6, 9, 1, 3, 7, 5, 8, 2]]) ; > GrabCycle:=proc(P,i) local C,j: > > #We view the cycle belnging to i as a list, starting at i > C:=[i]: > > #We find where i points to, call it j > j:=P[i]: > > while j<>i do > #sooner or later we will wind back at i (since the permutation is finite) so we append the new > #arrivals to the list C and rename j > C:=[op(C),j]: > j:=P[j]: > od: > > #We look at the place where it it is max > j:=max[index](C): > > #we rewrite the cycle with the largest entry first > > [op(j..nops(C),C),op(1..j-1,C)]: > > end: > PtoC:=proc(P) local n,StillToDo,L,i,C,L1,T: > > n:=nops(P): > > StillToDo:={seq(i,i=1..n)}: > > #L is a dynamically construcete list of cycles in the permutatib P, we start with the empty list > #until no one is left > L:=[]: > > while StillToDo<>{} do > #we pick the smallest survivor > i:=min(op(StillToDo)): > #we grab its cycle > C:=GrabCycle(P,i): > > # We append C to L > L:=[op(L),C]: > > #We update StillToDo by kicking out the members of C > > StillToDo:= StillToDo minus convert(C,set): > od: > > #So far L is a list of cycles but arranged in a random order > #Now we arrange them according to the convention that the first (largest) entries are increasing > #L1 is the list of first (largest) entries of each cycle: > L1:=[seq(L[i][1],i=1..nops(L))]: > > #We now make a table where T[a] is the cycle that starts with a > for i from 1 to nops(L) do > T[L[i][1]]:=L[i]: > od: > > #Now we sort L1 > L1:=sort(L1): > > #Now we rewrite L with the above convention > [seq(T[L1[i]],i=1..nops(L1))]: > > end: > > > #FunT(P): The fundamental transformation > FunT:=proc(P) local P1,i: > P1:=PtoC(P): > [seq(op(P1[i]),i=1..nops(P1))]: > end: > > #GenGp(S): inputs a pos. integer n and a set of permutations in {1,..n} outputs the subgroup generated by them > GG:=proc(S) local Gr,s,g,Gr1,Gr2: > Gr:=S: > Gr1:=Gr union {seq(seq(MulPers(s,g), s in S), g in Gr)}: > > while Gr<>Gr1 do > Gr2:=Gr1 union {seq(seq(MulPers(s,g), s in S), g in Gr1)}: > Gr:=Gr1: > Gr1:=Gr2: > od: > Gr1: > > end: > PtoC(P4); [[4, 1], [8], [9, 2, 6, 7, 5, 3]] ; > ; > P5:=randperm(9) Typesetting:-mprintslash([(P5 := [5, 7, 1, 4, 9, 8, 3, 6, 2])],[[5, 7, 1, 4, 9, 8, 3, 6, 2]]) ; > inv:=proc(pi) local n,i,j,co: > n:=nops(pi): > co:=0: > > for i from 1 to n do > for j from i+1 to n do > if pi[i]>pi[j] then > co:=co+1: > fi: > od: > od: > > co: > > end: > inv(P5) 20 ; > #7-5,7-3,7-6,7-2,7-4,7-1 ; > #5-3,5-2,5-4,5-2 ; > #3-2,3-1 ; > #6-2,6-4,6-1 ; > #8-2,8-4,8-1 ; > #2-1 ; > #9-4,901 ; > #4-1 ; > #Total: 22 ; > ; > maj:=proc(pi) local n,i,co: > n:=nops(pi): > co:=0: > > for i from 1 to n-1 do > if pi[i]>pi[i+1] then > co:=co+i: > fi: > od: > > > co: > > end: > maj(P5) 21 ; > #1+2+5+7+8=23 ; > ; > ; > #Q2. ; > # ; > # ; > # ; > InvGF:= proc(n,q) local L, num, i: > option remember: > > L:=permute(n): > num:= nops(L): > > add(q^inv(L[i]), i=1..num): > > end: > InvGF(3,q) q^3+2*q^2+2*q+1 ; > ; > MajGF:=proc(n,q) local L, num, i: > option remember: > > L:=permute(n): > num:= nops(L): > > add(q^maj(L[i]), i = 1..num): > > end: > MajGF(3,q); q^3+2*q^2+2*q+1 ; > seq(evalb(MajGF(n,q)=InvGF(n,q)), n=1..7) true, true, true, true, true, true, true ; > ; > ; > ; > #Q3. ; > simplify(InvGF(2,q)/InvGF(1,q)); 1+q ; > simplify(InvGF(3,q)/InvGF(2,q)); q^2+q+1 ; > simplify(InvGF(4,q)/InvGF(3,q)); q^3+q^2+q+1 ; > simplify(InvGF(5,q)/InvGF(4,q)); q^4+q^3+q^2+q+1 ; > #InvGF(n,q)/InvGF(n-1,q) = q^0+q^1+...+q^n-1 ; > #InvGF(n,q) = (q^0+q^1+...+q^n-1)*InvGF(n-1,q); > ; > ; > #Q4. ; > IsBad:= proc(P) local n, num, i, j, k: > option memeber: > > num:= nops(P): > for i from 3 to num do > for j from 1 to i do > if P[j] for k from 1 to j do > if P[k] < P[j] then > RETURN(True): > fi: > od: > fi: > od:od: > RETURN("not found"): > end: > IsBad([3,1,4,2,5]); True ; > IsBad([4,3,2,1,5]) "not found" ; > ABC:=proc(n) local num, L, i, S: > option remember: > S:={}: > L:=permute(n); > num := nops(L); > for i from 1 to num do > if evalb(IsBad(L[i])="not found") then > S:= S union {L[i]}: > fi: > od: > S: > end: > ABC(5) {[1, 5, 4, 3, 2], [2, 1, 5, 4, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3], [2, 5, 4, 3, 1], [3, 1, 5, 4, 2], [3, 2, 1, 5, 4], [3, 2, 5, 1, 4], [3, 2, 5, 4, 1], [3, 5, 1, 4, 2], [3, 5, 2, 1, 4], [3, 5, 2, 4, 1], [3, 5, 4, 1, 2], [3, 5, 4, 2, 1] , [4, 1, 5, 3, 2], [4, 2, 1, 5, 3], [4, 2, 5, 1, 3], [4, 2, 5, 3, 1], [4, 3, 1, 5, 2], [4, 3, 2, 1, 5], [4, 3, 2, 5, 1], [4, 3, 5, 1, 2], [4, 3, 5, 2, 1], [4, 5, 1, 3, 2], [4, 5, 2, 1, 3], [4, 5, 2, 3, 1], [4, 5, 3, 1, 2], [4, 5, 3, 2, 1] , [5, 1, 4, 3, 2], [5, 2, 1, 4, 3], [5, 2, 4, 1, 3], [5, 2, 4, 3, 1], [5, 3, 1, 4, 2], [5, 3, 2, 1, 4], [5, 3, 2, 4, 1], [5, 3, 4, 1, 2], [5, 3, 4, 2, 1], [5, 4, 1, 3, 2], [5, 4, 2, 1, 3], [5, 4, 2, 3, 1], [5, 4, 3, 1, 2], [5, 4, 3, 2, 1] } ; > ABC(8) {[1, 8, 7, 6, 5, 4, 3, 2], [2, 1, 8, 7, 6, 5, 4, 3], [2, 8, 1, 7, 6, 5, 4, 3], [2, 8, 7, 1, 6, 5, 4, 3], [2, 8, 7, 6, 1, 5, 4, 3], [2, 8, 7, 6, 5, 1, 4, 3], [ 2, 8, 7, 6, 5, 4, 1, 3], [2, 8, 7, 6, 5, 4, 3, 1], [3, 1, 8, 7, 6, 5, 4, 2], [3 , 2, 1, 8, 7, 6, 5, 4], [3, 2, 8, 1, 7, 6, 5, 4], [3, 2, 8, 7, 1, 6, 5, 4], [3, 2, 8, 7, 6, 1, 5, 4], [3, 2, 8, 7, 6, 5, 1, 4], [3, 2, 8, 7, 6, 5, 4, 1], [3, 8 , 1, 7, 6, 5, 4, 2], [3, 8, 2, 1, 7, 6, 5, 4], [3, 8, 2, 7, 1, 6, 5, 4], [3, 8, 2, 7, 6, 1, 5, 4], [3, 8, 2, 7, 6, 5, 1, 4], [3, 8, 2, 7, 6, 5, 4, 1], [3, 8, 7 , 1, 6, 5, 4, 2], [3, 8, 7, 2, 1, 6, 5, 4], [3, 8, 7, 2, 6, 1, 5, 4], [3, 8, 7, 2, 6, 5, 1, 4], [3, 8, 7, 2, 6, 5, 4, 1], [3, 8, 7, 6, 1, 5, 4, 2], [3, 8, 7, 6 , 2, 1, 5, 4], [3, 8, 7, 6, 2, 5, 1, 4], [3, 8, 7, 6, 2, 5, 4, 1], [3, 8, 7, 6, 5, 1, 4, 2], [3, 8, 7, 6, 5, 2, 1, 4], [3, 8, 7, 6, 5, 2, 4, 1], [3, 8, 7, 6, 5 , 4, 1, 2], [3, 8, 7, 6, 5, 4, 2, 1], [4, 1, 8, 7, 6, 5, 3, 2], [4, 2, 1, 8, 7, 6, 5, 3], [4, 2, 8, 1, 7, 6, 5, 3], [4, 2, 8, 7, 1, 6, 5, 3], [4, 2, 8, 7, 6, 1 , 5, 3], [4, 2, 8, 7, 6, 5, 1, 3], [4, 2, 8, 7, 6, 5, 3, 1], [4, 3, 1, 8, 7, 6, 5, 2], [4, 3, 2, 1, 8, 7, 6, 5], [4, 3, 2, 8, 1, 7, 6, 5], [4, 3, 2, 8, 7, 1, 6 , 5], [4, 3, 2, 8, 7, 6, 1, 5], [4, 3, 2, 8, 7, 6, 5, 1], [4, 3, 8, 1, 7, 6, 5, 2], [4, 3, 8, 2, 1, 7, 6, 5], [4, 3, 8, 2, 7, 1, 6, 5], [4, 3, 8, 2, 7, 6, 1, 5 ], [4, 3, 8, 2, 7, 6, 5, 1], [4, 3, 8, 7, 1, 6, 5, 2], [4, 3, 8, 7, 2, 1, 6, 5] , [4, 3, 8, 7, 2, 6, 1, 5], [4, 3, 8, 7, 2, 6, 5, 1], [4, 3, 8, 7, 6, 1, 5, 2], [4, 3, 8, 7, 6, 2, 1, 5], [4, 3, 8, 7, 6, 2, 5, 1], [4, 3, 8, 7, 6, 5, 1, 2], [ 4, 3, 8, 7, 6, 5, 2, 1], [4, 8, 1, 7, 6, 5, 3, 2], [4, 8, 2, 1, 7, 6, 5, 3], [4 , 8, 2, 7, 1, 6, 5, 3], [4, 8, 2, 7, 6, 1, 5, 3], [4, 8, 2, 7, 6, 5, 1, 3], [4, 8, 2, 7, 6, 5, 3, 1], [4, 8, 3, 1, 7, 6, 5, 2], [4, 8, 3, 2, 1, 7, 6, 5], [4, 8 , 3, 2, 7, 1, 6, 5], [4, 8, 3, 2, 7, 6, 1, 5], [4, 8, 3, 2, 7, 6, 5, 1], [4, 8, 3, 7, 1, 6, 5, 2], [4, 8, 3, 7, 2, 1, 6, 5], [4, 8, 3, 7, 2, 6, 1, 5], [4, 8, 3 , 7, 2, 6, 5, 1], [4, 8, 3, 7, 6, 1, 5, 2], [4, 8, 3, 7, 6, 2, 1, 5], [4, 8, 3, 7, 6, 2, 5, 1], [4, 8, 3, 7, 6, 5, 1, 2], [4, 8, 3, 7, 6, 5, 2, 1], [4, 8, 7, 1 , 6, 5, 3, 2], [4, 8, 7, 2, 1, 6, 5, 3], [4, 8, 7, 2, 6, 1, 5, 3], [4, 8, 7, 2, 6, 5, 1, 3], [4, 8, 7, 2, 6, 5, 3, 1], [4, 8, 7, 3, 1, 6, 5, 2], [4, 8, 7, 3, 2 , 1, 6, 5], [4, 8, 7, 3, 2, 6, 1, 5], [4, 8, 7, 3, 2, 6, 5, 1], [4, 8, 7, 3, 6, 1, 5, 2], [4, 8, 7, 3, 6, 2, 1, 5], [4, 8, 7, 3, 6, 2, 5, 1], [4, 8, 7, 3, 6, 5 , 1, 2], [4, 8, 7, 3, 6, 5, 2, 1], [4, 8, 7, 6, 1, 5, 3, 2], [4, 8, 7, 6, 2, 1, 5, 3], [4, 8, 7, 6, 2, 5, 1, 3], [4, 8, 7, 6, 2, 5, 3, 1], [4, 8, 7, 6, 3, 1, 5 , 2], [4, 8, 7, 6, 3, 2, 1, 5], [4, 8, 7, 6, 3, 2, 5, 1], [4, 8, 7, 6, 3, 5, 1, 2], [4, 8, 7, 6, 3, 5, 2, 1], [4, 8, 7, 6, 5, 1, 3, 2], [4, 8, 7, 6, 5, 2, 1, 3 ], [4, 8, 7, 6, 5, 2, 3, 1], [4, 8, 7, 6, 5, 3, 1, 2], [4, 8, 7, 6, 5, 3, 2, 1] , [5, 1, 8, 7, 6, 4, 3, 2], [5, 2, 1, 8, 7, 6, 4, 3], [5, 2, 8, 1, 7, 6, 4, 3], [5, 2, 8, 7, 1, 6, 4, 3], [5, 2, 8, 7, 6, 1, 4, 3], [5, 2, 8, 7, 6, 4, 1, 3], [ 5, 2, 8, 7, 6, 4, 3, 1], [5, 3, 1, 8, 7, 6, 4, 2], [5, 3, 2, 1, 8, 7, 6, 4], [5 , 3, 2, 8, 1, 7, 6, 4], [5, 3, 2, 8, 7, 1, 6, 4], [5, 3, 2, 8, 7, 6, 1, 4], [5, 3, 2, 8, 7, 6, 4, 1], [5, 3, 8, 1, 7, 6, 4, 2], [5, 3, 8, 2, 1, 7, 6, 4], [5, 3 , 8, 2, 7, 1, 6, 4], [5, 3, 8, 2, 7, 6, 1, 4], [5, 3, 8, 2, 7, 6, 4, 1], [5, 3, 8, 7, 1, 6, 4, 2], [5, 3, 8, 7, 2, 1, 6, 4], [5, 3, 8, 7, 2, 6, 1, 4], [5, 3, 8 , 7, 2, 6, 4, 1], [5, 3, 8, 7, 6, 1, 4, 2], [5, 3, 8, 7, 6, 2, 1, 4], [5, 3, 8, 7, 6, 2, 4, 1], [5, 3, 8, 7, 6, 4, 1, 2], [5, 3, 8, 7, 6, 4, 2, 1], [5, 4, 1, 8 , 7, 6, 3, 2], [5, 4, 2, 1, 8, 7, 6, 3], [5, 4, 2, 8, 1, 7, 6, 3], [5, 4, 2, 8, 7, 1, 6, 3], [5, 4, 2, 8, 7, 6, 1, 3], [5, 4, 2, 8, 7, 6, 3, 1], [5, 4, 3, 1, 8 , 7, 6, 2], [5, 4, 3, 2, 1, 8, 7, 6], [5, 4, 3, 2, 8, 1, 7, 6], [5, 4, 3, 2, 8, 7, 1, 6], [5, 4, 3, 2, 8, 7, 6, 1], [5, 4, 3, 8, 1, 7, 6, 2], [5, 4, 3, 8, 2, 1 , 7, 6], [5, 4, 3, 8, 2, 7, 1, 6], [5, 4, 3, 8, 2, 7, 6, 1], [5, 4, 3, 8, 7, 1, 6, 2], [5, 4, 3, 8, 7, 2, 1, 6], [5, 4, 3, 8, 7, 2, 6, 1], [5, 4, 3, 8, 7, 6, 1 , 2], [5, 4, 3, 8, 7, 6, 2, 1], [5, 4, 8, 1, 7, 6, 3, 2], [5, 4, 8, 2, 1, 7, 6, 3], [5, 4, 8, 2, 7, 1, 6, 3], [5, 4, 8, 2, 7, 6, 1, 3], [5, 4, 8, 2, 7, 6, 3, 1 ], [5, 4, 8, 3, 1, 7, 6, 2], [5, 4, 8, 3, 2, 1, 7, 6], [5, 4, 8, 3, 2, 7, 1, 6] , [5, 4, 8, 3, 2, 7, 6, 1], [5, 4, 8, 3, 7, 1, 6, 2], [5, 4, 8, 3, 7, 2, 1, 6], [5, 4, 8, 3, 7, 2, 6, 1], [5, 4, 8, 3, 7, 6, 1, 2], [5, 4, 8, 3, 7, 6, 2, 1], [ 5, 4, 8, 7, 1, 6, 3, 2], [5, 4, 8, 7, 2, 1, 6, 3], [5, 4, 8, 7, 2, 6, 1, 3], [5 , 4, 8, 7, 2, 6, 3, 1], [5, 4, 8, 7, 3, 1, 6, 2], [5, 4, 8, 7, 3, 2, 1, 6], [5, 4, 8, 7, 3, 2, 6, 1], [5, 4, 8, 7, 3, 6, 1, 2], [5, 4, 8, 7, 3, 6, 2, 1], [5, 4 , 8, 7, 6, 1, 3, 2], [5, 4, 8, 7, 6, 2, 1, 3], [5, 4, 8, 7, 6, 2, 3, 1], [5, 4, 8, 7, 6, 3, 1, 2], [5, 4, 8, 7, 6, 3, 2, 1], [5, 8, 1, 7, 6, 4, 3, 2], [5, 8, 2 , 1, 7, 6, 4, 3], [5, 8, 2, 7, 1, 6, 4, 3], [5, 8, 2, 7, 6, 1, 4, 3], [5, 8, 2, 7, 6, 4, 1, 3], [5, 8, 2, 7, 6, 4, 3, 1], [5, 8, 3, 1, 7, 6, 4, 2], [5, 8, 3, 2 , 1, 7, 6, 4], [5, 8, 3, 2, 7, 1, 6, 4], [5, 8, 3, 2, 7, 6, 1, 4], [5, 8, 3, 2, 7, 6, 4, 1], [5, 8, 3, 7, 1, 6, 4, 2], [5, 8, 3, 7, 2, 1, 6, 4], [5, 8, 3, 7, 2 , 6, 1, 4], [5, 8, 3, 7, 2, 6, 4, 1], [5, 8, 3, 7, 6, 1, 4, 2], [5, 8, 3, 7, 6, 2, 1, 4], [5, 8, 3, 7, 6, 2, 4, 1], [5, 8, 3, 7, 6, 4, 1, 2], [5, 8, 3, 7, 6, 4 , 2, 1], [5, 8, 4, 1, 7, 6, 3, 2], [5, 8, 4, 2, 1, 7, 6, 3], [5, 8, 4, 2, 7, 1, 6, 3], [5, 8, 4, 2, 7, 6, 1, 3], [5, 8, 4, 2, 7, 6, 3, 1], [5, 8, 4, 3, 1, 7, 6 , 2], [5, 8, 4, 3, 2, 1, 7, 6], [5, 8, 4, 3, 2, 7, 1, 6], [5, 8, 4, 3, 2, 7, 6, 1], [5, 8, 4, 3, 7, 1, 6, 2], [5, 8, 4, 3, 7, 2, 1, 6], [5, 8, 4, 3, 7, 2, 6, 1 ], [5, 8, 4, 3, 7, 6, 1, 2], [5, 8, 4, 3, 7, 6, 2, 1], [5, 8, 4, 7, 1, 6, 3, 2] , [5, 8, 4, 7, 2, 1, 6, 3], [5, 8, 4, 7, 2, 6, 1, 3], [5, 8, 4, 7, 2, 6, 3, 1], [5, 8, 4, 7, 3, 1, 6, 2], [5, 8, 4, 7, 3, 2, 1, 6], [5, 8, 4, 7, 3, 2, 6, 1], [ 5, 8, 4, 7, 3, 6, 1, 2], [5, 8, 4, 7, 3, 6, 2, 1], [5, 8, 4, 7, 6, 1, 3, 2], [5 , 8, 4, 7, 6, 2, 1, 3], [5, 8, 4, 7, 6, 2, 3, 1], [5, 8, 4, 7, 6, 3, 1, 2], [5, 8, 4, 7, 6, 3, 2, 1], [5, 8, 7, 1, 6, 4, 3, 2], [5, 8, 7, 2, 1, 6, 4, 3], [5, 8 , 7, 2, 6, 1, 4, 3], [5, 8, 7, 2, 6, 4, 1, 3], [5, 8, 7, 2, 6, 4, 3, 1], [5, 8, 7, 3, 1, 6, 4, 2], [5, 8, 7, 3, 2, 1, 6, 4], [5, 8, 7, 3, 2, 6, 1, 4], [5, 8, 7 , 3, 2, 6, 4, 1], [5, 8, 7, 3, 6, 1, 4, 2], [5, 8, 7, 3, 6, 2, 1, 4], [5, 8, 7, 3, 6, 2, 4, 1], [5, 8, 7, 3, 6, 4, 1, 2], [5, 8, 7, 3, 6, 4, 2, 1], [5, 8, 7, 4 , 1, 6, 3, 2], [5, 8, 7, 4, 2, 1, 6, 3], [5, 8, 7, 4, 2, 6, 1, 3], [5, 8, 7, 4, 2, 6, 3, 1], [5, 8, 7, 4, 3, 1, 6, 2], [5, 8, 7, 4, 3, 2, 1, 6], [5, 8, 7, 4, 3 , 2, 6, 1], [5, 8, 7, 4, 3, 6, 1, 2], [5, 8, 7, 4, 3, 6, 2, 1], [5, 8, 7, 4, 6, 1, 3, 2], [5, 8, 7, 4, 6, 2, 1, 3], [5, 8, 7, 4, 6, 2, 3, 1], [5, 8, 7, 4, 6, 3 , 1, 2], [5, 8, 7, 4, 6, 3, 2, 1], [5, 8, 7, 6, 1, 4, 3, 2], [5, 8, 7, 6, 2, 1, 4, 3], [5, 8, 7, 6, 2, 4, 1, 3], [5, 8, 7, 6, 2, 4, 3, 1], [5, 8, 7, 6, 3, 1, 4 , 2], [5, 8, 7, 6, 3, 2, 1, 4], [5, 8, 7, 6, 3, 2, 4, 1], [5, 8, 7, 6, 3, 4, 1, 2], [5, 8, 7, 6, 3, 4, 2, 1], [5, 8, 7, 6, 4, 1, 3, 2], [5, 8, 7, 6, 4, 2, 1, 3 ], [5, 8, 7, 6, 4, 2, 3, 1], 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7, 4, 6, 5, 2, 1, 3], [8, 7, 4, 6, 5, 2, 3, 1], [8, 7, 4, 6, 5, 3, 1, 2], [8, 7, 4, 6, 5, 3, 2, 1], [8, 7, 5, 1, 6, 4, 3, 2], [8, 7, 5, 2 , 1, 6, 4, 3], [8, 7, 5, 2, 6, 1, 4, 3], [8, 7, 5, 2, 6, 4, 1, 3], [8, 7, 5, 2, 6, 4, 3, 1], [8, 7, 5, 3, 1, 6, 4, 2], [8, 7, 5, 3, 2, 1, 6, 4], [8, 7, 5, 3, 2 , 6, 1, 4], [8, 7, 5, 3, 2, 6, 4, 1], [8, 7, 5, 3, 6, 1, 4, 2], [8, 7, 5, 3, 6, 2, 1, 4], [8, 7, 5, 3, 6, 2, 4, 1], [8, 7, 5, 3, 6, 4, 1, 2], [8, 7, 5, 3, 6, 4 , 2, 1], [8, 7, 5, 4, 1, 6, 3, 2], [8, 7, 5, 4, 2, 1, 6, 3], [8, 7, 5, 4, 2, 6, 1, 3], [8, 7, 5, 4, 2, 6, 3, 1], [8, 7, 5, 4, 3, 1, 6, 2], [8, 7, 5, 4, 3, 2, 1 , 6], [8, 7, 5, 4, 3, 2, 6, 1], [8, 7, 5, 4, 3, 6, 1, 2], [8, 7, 5, 4, 3, 6, 2, 1], [8, 7, 5, 4, 6, 1, 3, 2], [8, 7, 5, 4, 6, 2, 1, 3], [8, 7, 5, 4, 6, 2, 3, 1 ], [8, 7, 5, 4, 6, 3, 1, 2], [8, 7, 5, 4, 6, 3, 2, 1], [8, 7, 5, 6, 1, 4, 3, 2] , [8, 7, 5, 6, 2, 1, 4, 3], [8, 7, 5, 6, 2, 4, 1, 3], [8, 7, 5, 6, 2, 4, 3, 1], [8, 7, 5, 6, 3, 1, 4, 2], [8, 7, 5, 6, 3, 2, 1, 4], [8, 7, 5, 6, 3, 2, 4, 1], [ 8, 7, 5, 6, 3, 4, 1, 2], [8, 7, 5, 6, 3, 4, 2, 1], [8, 7, 5, 6, 4, 1, 3, 2], [8 , 7, 5, 6, 4, 2, 1, 3], [8, 7, 5, 6, 4, 2, 3, 1], [8, 7, 5, 6, 4, 3, 1, 2], [8, 7, 5, 6, 4, 3, 2, 1], [8, 7, 6, 1, 5, 4, 3, 2], [8, 7, 6, 2, 1, 5, 4, 3], [8, 7 , 6, 2, 5, 1, 4, 3], [8, 7, 6, 2, 5, 4, 1, 3], [8, 7, 6, 2, 5, 4, 3, 1], [8, 7, 6, 3, 1, 5, 4, 2], [8, 7, 6, 3, 2, 1, 5, 4], [8, 7, 6, 3, 2, 5, 1, 4], [8, 7, 6 , 3, 2, 5, 4, 1], [8, 7, 6, 3, 5, 1, 4, 2], [8, 7, 6, 3, 5, 2, 1, 4], [8, 7, 6, 3, 5, 2, 4, 1], [8, 7, 6, 3, 5, 4, 1, 2], [8, 7, 6, 3, 5, 4, 2, 1], [8, 7, 6, 4 , 1, 5, 3, 2], [8, 7, 6, 4, 2, 1, 5, 3], [8, 7, 6, 4, 2, 5, 1, 3], [8, 7, 6, 4, 2, 5, 3, 1], [8, 7, 6, 4, 3, 1, 5, 2], [8, 7, 6, 4, 3, 2, 1, 5], [8, 7, 6, 4, 3 , 2, 5, 1], [8, 7, 6, 4, 3, 5, 1, 2], [8, 7, 6, 4, 3, 5, 2, 1], [8, 7, 6, 4, 5, 1, 3, 2], [8, 7, 6, 4, 5, 2, 1, 3], [8, 7, 6, 4, 5, 2, 3, 1], [8, 7, 6, 4, 5, 3 , 1, 2], [8, 7, 6, 4, 5, 3, 2, 1], [8, 7, 6, 5, 1, 4, 3, 2], [8, 7, 6, 5, 2, 1, 4, 3], [8, 7, 6, 5, 2, 4, 1, 3], [8, 7, 6, 5, 2, 4, 3, 1], [8, 7, 6, 5, 3, 1, 4 , 2], [8, 7, 6, 5, 3, 2, 1, 4], [8, 7, 6, 5, 3, 2, 4, 1], [8, 7, 6, 5, 3, 4, 1, 2], [8, 7, 6, 5, 3, 4, 2, 1], [8, 7, 6, 5, 4, 1, 3, 2], [8, 7, 6, 5, 4, 2, 1, 3 ], [8, 7, 6, 5, 4, 2, 3, 1], [8, 7, 6, 5, 4, 3, 1, 2], [8, 7, 6, 5, 4, 3, 2, 1] } ; > ; > ; > ; > ; > #Q5. ; > #FunT(P): The fundamental transformation > FunT:=proc(P) local P1,i: > P1:=PtoC(P): > [seq(op(P1[i]),i=1..nops(P1))]: > end: > PtoC:=proc(P) local n,StillToDo,L,i,C,L1,T: > > n:=nops(P): > > StillToDo:={seq(i,i=1..n)}: > > #L is a dynamically construcete list of cycles in the permutatib P, we start with the empty list > #until no one is left > L:=[]: > > while StillToDo<>{} do > #we pick the smallest survivor > i:=min(op(StillToDo)): > #we grab its cycle > C:=GrabCycle(P,i): > > # We append C to L > L:=[op(L),C]: > > #We update StillToDo by kicking out the members of C > > StillToDo:= StillToDo minus convert(C,set): > od: > > #So far L is a list of cycles but arranged in a random order > #Now we arrange them according to the convention that the first (largest) entries are increasing > #L1 is the list of first (largest) entries of each cycle: > L1:=[seq(L[i][1],i=1..nops(L))]: > > #We now make a table where T[a] is the cycle that starts with a > for i from 1 to nops(L) do > T[L[i][1]]:=L[i]: > od: > > #Now we sort L1 > L1:=sort(L1): > > #Now we rewrite L with the above convention > [seq(T[L1[i]],i=1..nops(L1))]: > > end: > FunT([5,1,3,2,4]) [3, 5, 4, 2, 1] ; > PtoC([5,1,3,2,4]) [[3], [5, 4, 2, 1]] ; > nops(%) 2 ; > CtoP:=proc(C) local i,L,n,T,C1,j: > L:=[seq(op(C[i]),i=1..nops(C))]: > n:=nops(L): > > if convert(L,set)<>{seq(i,i=1..n)} then > RETURN(FAIL): > fi: > > for i from 1 to nops(C) do > C1:=C[i]: > for j from 1 to nops(C1)-1 do > T[C1[j]]:=C1[j+1]: > od: > T[C1[nops(C1)]]:=C1[1]: > od: > > [seq(T[i],i=1..n)]: > end: > InvFunT:= proc(P) local P1, i: > option remember: > > > P1:=PtoC(P): > [seq(op(P1[i]),i=1..nops(P1))]: > > > > > end: > ; > ; > ; > ;