Factoring out permutations of SN

I'm trying to remove certain permutations from S_N.

I want to remove all permutations that leave the first [0...k[ indices invariant (except for Identity element). So what I'm looking for is S_{1,.., N} / S_{k, .., N}.

I understand that I could do ( N= 10, k=5 )

SN = SymmetricGroup(range(10))
SkN = SymmetricGroup(range(5,10)
permutations = [sigma for sigma in SN if sigma not in SkN]


but this is extremely slow. My goal here is to speed up my code and I end up generating twice as much permutations and doing extreme list checking.

I'm pretty sure people were smart enough to implement a feature that gives me back a generator Squotient = SN/SkN, but I could not find this in the documentation. Any ideas?

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Please describe the needed object, and its type, and its further use. So far, one can give as answer for the one or the other line as follows:

(1) The list S in

sage: K, N = 3, 6    # other values
sage: S = [s for s in SymmetricGroup(range(N)) if [s(k) for k in [0..K-1]] != [0..K-1] ]
sage: len(S) == factorial(N) - factorial(K)
True


so that it must be also stored as a list.

(2) The left/right quotient in

sage: len( G.cosets(H, side='right') ), len( G.cosets(H, side='right') )
(120, 120)


Stored again.

(3) A generator for the one or the other list.

(4) A "programming device" to loop over SN and do something for some $s$ if not in SkN, e.g.

for s in SN:
if [s(k) for k in [0..K-1]] == [0..K-1]:
continue
# real work

( 2018-03-14 20:41:43 +0100 )edit

Thank you, (2) is what I was looking for, but I expected it would not return a list and return an iterator instead. (1) and (4) seems the most appropriate.

( 2018-03-15 13:36:05 +0100 )edit