#C28.txt, May 5, 2025
Help28:=proc(): print(` Pri(pi,k,j), SUC(pi,k), EstSUC(n,k,K)  `):end:
with(combinat):
PriNuray:= proc(pi, k, j)  local c, i :
c:=1:
i:= j:

while pi[i] <> j and c <= k do
    c:= c+ 1:
    i:= pi[i]:
od:

if c <=  k then
 print(`SUCCESS!`):
 return true:

else
 print(`FAIL!`):
 return false:
fi:

end:

#From Nick
OneCycleEGF := proc(n, x)
    local k;
    add(x^k/k, k = 1 .. n);
end proc:


ExpCoeffs := proc(F, n, x)
    local S, k;
    S := series(exp(F), x = 0, n+1);
    [seq(coeff(S, x, k) * k!  , k = 0 .. n)];
end proc:

F:=OneCycleEGF(50,x):
cof:=ExpCoeffs(F,100,x):
number_of_permutations:=cof[101]: #number of permutations with only cycles length <= 50
p:=number_of_permutations / (100!):

print(evalf(p));
#end from Nick

# Currently set to spit out true or false. You can have it return k0 to see how many attempts it actually took.
#By Omar Garcia
Pri:=proc(pi,k,j) local j0,k0:
j0 := pi[j]:
k0:= 1:
while (k0 <= k) and (j0 <> j) do
j0 := pi[j0]:
k0+=1:
od:
if k0 > k then return(false): else return(true): fi:
end:

PriAurora := proc(pi,k,j) local g,c:
g := 0:         # number of guesses
c := pi[j]:     # current locker

while g < k do  
    if c = j then
        RETURN(TRUE)
    fi:

    c := pi[c]:
    g++:
od:

FAIL:
end:

#SUC(pi,k): for a permutation pi representing nops(pi) prisoners, and each allowed to look at k boxes,
#does it succeed
SUC:=proc(pi,k) local j:
if coeff(add(Pri(pi, k, j), j=1..nops(pi)),true) = nops(pi) then return(true): else return(false): fi:
end: 

#Code by Salman Manzoor
SUCsalmans:=proc(pi,k) local i:
for i from 1 to nops(pi) do
if not Pri(pi,k,i) then RETURN(false): fi:
od:
true:
end:

#EstSUC(n,k,K), estimates the prob. of success with n prisoners and k tries doing it K times
EstSUC:=proc(n,k,K) local i:
evalf(coeff(add(SUC(randperm(n),k),i=1..K),true)/K):
end: