Abstract
We present a finite-order system of recurrence relations for the permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for and 3) and 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 -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.
Highlights
Significance and Complexity of Circulant Permanents The permanent, per C, and the determinant, det C, of a n × n matrix C correspond to two major operations—the symmetrization and the anti-symmetrization, respectively
The permanents are well-known in mathematical physics, especially in quantum computing science and the quantum field theory of interacting Bose fields [1,2,3,4,5,6,7,8]
The permanents have been studied in mathematics for more than a century, the most actively after discovery of the Ryser’s algorithm [19], the publication of the comprehensive book “Permanents” [13], proof of the famous Valiant’s theorem stating that their computing is a P-hard problem within the computational complexity theory [20] and a recent development of a fully polynomial randomized approximation scheme [21,22] for their computing
Summary
Significance and Complexity of Circulant Permanents The permanent, per C, and the determinant, det C, of a n × n matrix C correspond to two major operations—the symmetrization and the anti-symmetrization, respectively. One specific goal is to demonstrate practical ways of this method and communicate the new results on finding the system of recurrence relations for calculating nontrivial, multiparametric circulant permanents with a band of k = 1, 2 or 3 different diagonals. Especially of matrices whose entries are continuous variables, the proposed method of the recursion of permanents with defects is more powerful and Entropy 2021, 23, 1423 efficient than a widely known method of rook polynomials [17,51,52,53,54,57] The former incorporates all entire analytic, algebraic and combinatorial information on the permanental polynomial of k variables. M(i1i2|j1j2) is the (n − 2) × (n − 2) submatrix obtained from a n × n matrix M after deleting rows i1 and i2, and columns j1 and j2
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
