# Permutations & Polynomials

zak seidov zakseidov at yahoo.com
Sat Apr 1 23:19:19 CEST 2006

Consider n! permutations of n (n=4 in this case)
integers. First permutation is per(1)={1,2,3,4} and
the last one is per(24)={4,3,2,1}. For per(1) we have
a(i)=i, and for per(24), a(i)=5-i. What about other
cases?
In general, a(i)=A+B*i +C*i^2+D*i^3, and we can find
coefficients of polynomials for all 24 permutations:
{A,B,C,D}={{0,1,0,0},{5,-(47/6),9/2,-(2/3)},{-10,35/2,-(15/2),1},{0,-(1/6),3/2,-(1/3)},{-15,151/6,-(21/2),4/3},{-10,49/3,-6,2/3},{10,-(77/6),11/2,-(2/3)},{15,-(65/3),10,-(4/3)},{-10,121/6,-(19/2),4/3},{5,-(19/3),4,-(2/3)},{-15,167/6,-(25/2),5/3},{-5,61/6,-(7/2),1/3},{10,-(61/6),7/2,-(1/3)},{20,-(167/6),25/2,-(5/3)},{0,19/3,-4,2/3},{15,-(121/6),19/2,-(4/3)},{-10,65/3,-10,4/3},{-5,77/6,-(11/2),2/3},{15,-(49/3),6,-(2/3)},{20,-(151/6),21/2,-(4/3)},{5,1/6,-(3/2),1/3},{15,-(35/2),15/2,-1},{0,47/6,-(9/2),2/3},{5,-1,0,0}},
or multiplying by 6 we have the sequence integers (not
in OEIS?):
{0,6,0,0,30,-47,27,-4,-60,105,-45,6,0,-1,9,-2,-90,151,-63,8,-60,98,-36,4,60,-77,33,-4,90,-130,60,-8,-60,121,-57,8,30,-38,24,-4,-90,167,-75,10,-30,61,-21,2,60,-61,21,-2,120,-167,75,-10,0,38,-24,4,90,-121,57,-8,-60,130,-60,8,-30,77,-33,4,90,-98,36,-4,120,-151,63,-8,30,1,-9,2,90,-105,45,-6,0,47,-27,4,30,-6,0,0}.
It's of interest to check this polynomials for primes
etc. (if they aint known already!).
Also it's not difficult to find (coefficients of)
these polynomials for other n's.
Thanks, Zak

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