Help23:=proc(): print(` GrabCycle(S,T), LtoC(L), CtoL(C), FindCycle(f,i) , RandF(n), CanC(C) , Joyal(f), RandTree(n)  `): end:

#RandF(n): A random function {1,..n}->{1,..n}
RandF:=proc(n) local ra,i:
ra:=rand(1..n):
[seq(ra(),i=1..n)]:
end:

#GrabCycle(S,T): Given a set S of vertices and a table T grabs a cycle that belongs to the first element of S
GrabCycle:=proc(S,T) local s,cu,ne,C:
s:=S[1]:
C:=[]:
cu:=s:
ne:=T[cu]:

if ne=s then
RETURN([s]):
fi:

while ne<>s do
C:=[op(C),cu]:
 ne:=T[cu]:
cu:=ne:
od:

C:

end:




#LtoC(L): Given a permutation of an arbitary set of positive integers in one-line notation
#converts it to a collection of cycles. Try:
#LtoC([6,1,4,5]);
LtoC:=proc(L)  local T,L1,C,S,C1,i:
L1:=sort(L):

for i from 1 to nops(L) do
T[L1[i]]:=L[i]:
od:

#S  is the vertices yet to be visited
S:={op(L)}:

#C is the set of cycles

C:={}:

while S<>{} do
 C1:=GrabCycle(S,op(T)):
 C:=C union {C1}:
 S:=S minus {op(C1)}:
od:

C:

end:


#CtoL(C): Given a collection of cycles, converts it to one line notation
CtoL:=proc(C) local j,C1,S, T,i:

S:=[]:
for C1 in C do
 for j from 1 to nops(C1)-1 do
 T[C1[j]]:=C1[j+1]:
 od:
T[C1[nops(C1)]]:=C1[1]:
S:=[op(S),op(C1)]:
od:
S:=sort(S):
[seq(T[S[i]],i=1..nops(S))]:
end:




#CanC(C): The canonical form of the cycle C
CanC:=proc(C) local s,i:
s:=min(op(C)):

for i from 1 to nops(C) while C[i]<>s do od:
[op(i..nops(C),C),op(1..i-1,C)]:
end:

#FindCycle(f,i): inputa a function f from {1, ...n} to {1,...,n} and a vertex i , 1<=i<=n output
#the cycle it leads to, and where it hits it. Try:
#FindCycle([1,2,2,2,3],3);
FindCycle:=proc(f,i) local n,cu,ne,C,j:
n:=nops(f):
if not 1<=i and i<=n then
 RETURN(FAIL):
fi:

cu:=i:

C:=[]:

while not member(cu,C) do
 C:=[op(C),cu]:
 ne:=f[cu]:
 cu:=ne:
od:

C:=[op(C),ne]:

for j from 1 to nops(C) while C[j]<>C[nops(C)] do od:
CanC([op(j+1..nops(C),C)]),C[nops(C)]:
end:


#Joyal(f): The joyal mapping. Inputs a function f from {1,..,n} to {1, ..n} outputs
# a doubly-rooted tree
Joyal:=proc(f) local n,C,i,T,S,j,s,L:

n:=nops(f):


C:={seq(FindCycle(f,i)[1],i=1..n)}:

S:={seq(op(C[i]),i=1..nops(C))}:

for s in S do
T[s]:={}:
od:

for i from 1 to n do
 if not member(i,S) then
  j:=FindCycle(f,i)[2]:
  T[j]:=T[j] union {{i,f[i]}}:
 fi:
od:


L:=CtoL(C):

[{seq(op(T[L[i]]),i=1..nops(L))} union {seq({L[i],L[i+1]},i=1..nops(L)-1)}, [L[1],L[nops(L)] ] ]:
 
end:


#RandTree(n): A random tree on n vertices using the Joyal bijection. Try:
#RandTree(30);
RandTree:=proc(n) local f:
f:=RandF(n):
Joyal(f)[1]:
end: