Geodesic
The number of terms in the permanent and the determinant of a generic circulant matrix
Journal of Algebraic Combinatorics, Tome 20 (2004) no. 1, pp. 55-60.

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Summary: Let $A = ( a _{ij})$ be the generic $n \times n$ circulant matrix given by $a _{ij} = x _{ i + j }$, with subscripts on $x$ interpreted mod $n$. Define $d( n)$ (resp. $p( n))$ to be the number of terms in the determinant (resp. permanent) of $A$. The function $p( n)$ is well-known and has several combinatorial interpretations. The function $d( n)$, on the other hand, has not been studied previously. We show that when $n$ is a prime power, $d( n) = p( n)$.
Keywords: generic circulant matrix, determinant, permanent 
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Thomas, Hugh. The number of terms in the permanent and the determinant of a generic circulant matrix. Journal of Algebraic Combinatorics, Tome 20 (2004) no. 1, pp. 55-60. http://geodesic.mathdoc.fr/item/JAC_2004__20_1_a3/