A084575 Number of terms in polynomial expression for determinant of generic circulant matrix of order n.

1, 2, 4, 10, 26, 68, 246, 810, 2704, 7492, 32066, 86500, 400024, 1366500, 4614524, 18784170, 68635478

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 13 2003

Keywords

• nonn
• hard
• more

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).

Examples

			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.
		

Links

• Hugh Thomas, The number of terms in the permanent ..., arXiv:math/0301048 [math.CO], 2003.

Crossrefs

Cf. A003239.

Programs

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 *)

Formula

a(n) <= A003239(n), with = if n is a prime power. For other values of n little is known.

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