Is there any algebraic structure that elegantly generates *all* permutations? I'm not exactly sure what you mean by algebraic structure, but "base factorial" gives an isomorphism between natural numbers and permutations.
https://en.wikipedia.org/wiki/Factorial_number_system On Fri, Mar 18, 2016 at 11:31 AM, Henry Baker <hbaker1@pipeline.com> wrote:
Answering own question:
Lagrange interpolation can allow a polynomial to construct *any* function on a finite field(?), yes/no?
So such an interpolation should allow any 1-1 function.
Is there any elegant form of this interpolation that is specialized to 1-1 functions?
At 11:08 AM 3/18/2016, Henry Baker wrote:
A cyclic group has a generator that generates all of the elements of the group; i.e., it produces a permutation of the elements of the group.
Different generators may produce different permutations.
Is there any algebraic structure that elegantly generates *all* permutations?
One problem with generators is that there aren't nearly enough of them: there's only N elements, but N! permutations.
Are there any other mechanisms -- perhaps high-degree polynomial evaluation or matrices -- that can generate all (& only) permutations?
Obviously, matrices can permute the elements of rows & columns, but I'm interested in 1-1 functions of the elements themselves.
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