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I modified Max Alekseyev's answer. The result of the following codes agree with the result of the function LongestPerm in the post.

def LongestPermWInSBWInverseSA(W,A,B):
t1=StablizerOfTuple(A)
t2=StablizerOfTuple(sorted(B))
s=W.simple_reflections()
#print(t1,t2)
t3=[]
for i in t1:
t3.append(s[i])
t4=[]
for i in t2:
t4.append(s[i])
t5=find_permutation2(B,sorted(B))
w=W.one()
for i in t5:
w=w*s[i]
#print(t4,w,t3)
r2=LongestPermInDoubleCosetWeylGroup(W,t4,w,t3)
r=r2

return r

def find_permutation2(L1, L2):
perm = Word(L1).standard_permutation() / Word(L2).standard_permutation()
assert [L2[i-1] for i in perm] == L1
r=perm.reduced_word()
return r

def LongestPermInDoubleCosetWeylGroup(W,S1,w,S2):
g1=W.subgroup(S1)
g2=W.subgroup(S2)
winner = W.one()
for u1 in g1:
for u2 in g2:
t1=u1*w*u2
if t1.length()>winner.length():
winner=t1
r=winner

return r

A=[1,1,2,3,3,4,6]
B=[12,9,12,15,15,14,12]
k = len(A)
W = WeylGroup('A'+str(k-1), prefix = 's')
t1=LongestPermWInSBWInverseSA(W,A,B)
print(t1)


I modified Max Alekseyev's answer. The result of the following codes agree with the result of the function LongestPerm in the post.

def LongestPermWInSBWInverseSA(W,A,B):
t1=StablizerOfTuple(A)
t2=StablizerOfTuple(sorted(B))
s=W.simple_reflections()
#print(t1,t2)
t3=[]
for i in t1:
t3.append(s[i])
t4=[]
for i in t2:
t4.append(s[i])
t5=find_permutation2(B,sorted(B))
w=W.one()
for i in t5:
w=w*s[i]
#print(t4,w,t3)
r2=LongestPermInDoubleCosetWeylGroup(W,t4,w,t3)
r=r2

return r

def find_permutation2(L1, L2):
perm = Word(L1).standard_permutation() / Word(L2).standard_permutation()
assert [L2[i-1] for i in perm] == L1
r=perm.reduced_word()
return r

def LongestPermInDoubleCosetWeylGroup(W,S1,w,S2):
g1=W.subgroup(S1)
g2=W.subgroup(S2)
winner = W.one()
for u1 in g1:
for u2 in g2:
t1=u1*w*u2
if t1.length()>winner.length():
winner=t1
r=winner

return r

def SymmetricGroupActionOnListSi(i,L): #  w=s_i,  L=[a1,a2,...,an]
r1=[]
for j in L:
r1.append(j)

t1=r1[i-1]
r1[i-1]=r1[i]
r1[i]=t1

r=r1

return r

def StablizerOfTuple(A): # A is a list, weakly increasing
k = len(A)
r=[]
for i in [1..k-1]:
t1=SymmetricGroupActionOnListSi(i,A)
if t1==A:
r.append(i)
return r

A=[1,1,2,3,3,4,6]
B=[12,9,12,15,15,14,12]
k = len(A)
W = WeylGroup('A'+str(k-1), prefix = 's')
t1=LongestPermWInSBWInverseSA(W,A,B)
print(t1)