OFFSET
1,2
COMMENTS
Define an n X n matrix A[i,j] by A[i,j]=x_(i+j), subscripts on x being interpreted mod n. This is a generic circulant matrix. If we expand det(A) we obtain a polynomial in the x_i. Define a(n) to be the number of terms in this polynomial after like terms have been combined. (Replacing det(A) with per(A), the permanent of A, we get sequence A003239).
LINKS
Hugh Thomas, The number of terms in the permanent ..., arXiv:math/0301048 [math.CO], 2003.
FORMULA
a(n) <= A003239(n), with = if n is a prime power. For other values of n little is known.
EXAMPLE
Example : for n=2 the matrix is
x2,x1
x1,x2
and the determinant is (x_2)^2 - (x_1)^2 so a(2) = 2 and likewise for the permanent.
MATHEMATICA
Table[Clear[x]; r=Array[x, n]; m=Table[RotateRight[r, i], {i, 0, n-1}]; Length[Expand[Det[m]]], {n, 10}] (* T. D. Noe, Oct 22 2008 *)
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 13 2003
EXTENSIONS
a(13) term added by T. D. Noe, Oct 22 2008
a(14) and a(15) from Roman Pearce, Aug 30 2014
a(16) and a(17) from Robert Israel, Aug 30 2014
STATUS
approved