Dokovic's conjecture : a survey of partial results



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Dokovic's conjecture states that the average permanental minor of a doubly stochastic matrix is never greater than the permanent. This conjecture is equivalent to the monotonicity of the permanent on line segments beginning at the center of doubly stochastic space. Partial results are presented and the methods used are discussed. The only complete proof, the case n=3, is also presented. Combinatorial arguments which prove validity on specific line segments are discussed and the following new result is proved by induction: the conjecture is valid on the line segment between any two permutation matrices. Matrix theory arguments which prove validity on specific subsets are discussed as well as a topological argument. The conclusion compares the relative possibilities of generalizing these arguments.



Permanents (Matrices)