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On the coefficients of the permanent and the determinant of a circulant matrix: applications. (English) Zbl 1407.15037

Summary: Let \( d(N)\) (resp., \( p(N)\)) be the number of summands in the determinant (resp., permanent) of an \( N\times N\) circulant matrix \( A=(a_{ij})\) given by \( a_{ij}=X_{i+j}\) where \( i+j\) should be considered \(\mathrm{mod}\,N\). This short note is devoted to proving that \( d(N)=p(N)\) if and only if \( N\) is a prime power. We then give an application to homogeneous monomial ideals failing the weak Lefschetz property.

MSC:

15B05 Toeplitz, Cauchy, and related matrices
15A15 Determinants, permanents, traces, other special matrix functions
13E10 Commutative Artinian rings and modules, finite-dimensional algebras

Software:

Macaulay2
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References:

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