Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Calculating the permanent of a (0, 1) matrix is a #P-complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most ll-known example is the fixed-jump (0, 1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation.#R##N##R##N#In this note  show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the first step is to show that the cycle-covers of directed circulant graphs can be evaluated using a transfer matrix argument. The second is to show that the associated transfer matrices, while very large, actually have much smaller characteristic polynomials than would a-priori be expected.#R##N##R##N#An important consequence of this new viewpoint is that, in combination with a new recursive decomposition of circulant-graphs, it permits extending Minc's result to calculating the permanent of the much larger class of circulant matrices with non-fixed (but linear) jumps.
['Mordecai J. Golin', 'Yiu Cho Leung', 'Yajun Wang']
I am writing a research paper on a idea Permanents of circulants: a transfer matrix approach
[{"from": "human", "value": "I am writing a research paper on a idea Permanents of circulants: a transfer matrix approach"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Calculating the permanent of a (0, 1) matrix is a #P-complete problem but there are some classes of structured matrices for which the permanent is calculable in polynomial time. The most ll-known example is the fixed-jump (0, 1) circulant matrix which, using algebraic techniques, was shown by Minc to satisfy a constant-coefficient fixed-order recurrence relation.#R##N##R##N#In this note  show how, by interpreting the problem as calculating the number of cycle-covers in a directed circulant graph, it is straightforward to reprove Minc's result using combinatorial methods. This is a two step process: the first step is to show that the cycle-covers of directed circulant graphs can be evaluated using a transfer matrix argument. The second is to show that the associated transfer matrices, while very large, actually have much smaller characteristic polynomials than would a-priori be expected.#R##N##R##N#An important consequence of this new viewpoint is that, in combination with a new recursive decomposition of circulant-graphs, it permits extending Minc's result to calculating the permanent of the much larger class of circulant matrices with non-fixed (but linear) jumps."}]