(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Question: How many simple substitution ciphers are there where no point is fixed (ie: no letter is mapped to itself)?

EDIT: Incase the termonoligy is different, a simple substitution cipher is a mapping where the plaintext in english is encoded so that every letter is mapped to a different letter.

IE:

a maps to Z

b maps to S

c maps to K

etcetcetc

thus abc plaintext is mapped to ZSK.

3. The attempt at a solution

So, I did a few examples to see if I could get some insight into the problem:

1 Letter = 0 mappings

2 Letters = 1 mapping

3 Letters = 2 mappings

4 Letters = 9 mappings

5 Letters = 44 mappings

I stopped here

While I found the reason for the complexity of determining this number I didn't come across any way to do it in general for 26 letters (for English). I know it has to be above (n-1)! and below n! for n letters but other then that I'm unsure.

I'm thinking I can just brute force it using algebraic notation, for example in the case of 4 letters I counted the mappings:

1 - 4cycle and with 4 letters there are 6 distinct combinations

2 - 2cycles and with 4 letters there are 3 distinct combinations

But, with 26 letters the number of different cycle combinations becomes cumbersome.

Any guidance would be appreciated.

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# Simple Substitution Ciphers

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