I wonder if your permutations are related to the ones described here: http://oeis.org/A027361/ On Sun, Jul 14, 2013 at 8:35 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Yep, you nailed it. All is forgiven.
--Dan
P.S. So, it turns out that this analysis leads to all but one permutation in S_4 -- and its inverse -- being realizable. Can anyone figure out which one this is?
On 2013-07-14, at 5:15 PM, Victor Miller wrote:
That's what I get for writing that while sitting by the swimming pool. So given a sequence of polynomials with f(x) = 0, we have a nested sequence of partitions: first partition them by the value of the first derivative. Within one such partition, if it has more than one member, partition it by the value of the second derivative, etc. The permutations that you get must be compatible with these partitions. If the degree which actually refines a sub-partition is even, it leaves the order the same when flipping signs, if it's odd, it reverses the order. There are only three relevant real values for the coefficients: >0, = 0 and < 0.
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