  We present a finite-order system of recurrence relations for a permanent of
circulant matrices containing a band of k any-value diagonals on top of a
uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such
recurrence relations which is based on the permanents of the matrices with
defects. The proposed system of linear recurrence equations with variable
coefficients provides a powerful tool for the analysis of the circulant
permanents, their fast, linear time computing and finding their asymptotics in
a large-matrix-size limit. The latter problem is an open fundamental problem.
Its solution would be tremendously important for a unified analysis of a wide
range of the nature's #P-hard problems, including problems in the physics of
many-body systems, critical phenomena, quantum computing, quantum field theory,
theory of chaos, fractals, theory of graphs, number theory, combinatorics,
cryptography, etc.
