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In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. [1] Both are special cases of a more general function of a matrix called the immanant.
The permanent is defined similarly to the determinant, as a sum of products of sets of matrix entries that lie in distinct rows and columns. However, where the determinant weights each of these products with a ±1 sign based on the parity of the set , the permanent weights them all with a +1 sign.
The computational complexity of the permanent also has some significance in other aspects of complexity theory: it is not known whether NC equals P (informally, whether every polynomially-solvable problem can be solved by a polylogarithmic-time parallel algorithm) and Ketan Mulmuley has suggested an approach to resolving this question that ...
The hafnian of a symmetric matrix is defined as = {,},, where is the set of all partitions of the set {,, …,} into subsets of size . [2] [3]This definition is similar to that of the Pfaffian, but differs in that the signatures of the permutations are not taken into account.
The permanent of a (0,1)-matrix is equal to the number of vertex-disjoint cycle covers of a directed graph with this adjacency matrix. This fact is used in a simplified proof showing that computing the permanent is #P-complete. [5]
The loss of the Senate by the Democratic Party may lead to a permanent Republican Senate, limiting the enactment of broad campaign promises, entrenching the perception of dysfunction in Washington ...
The permanent is therefore bounded by the product of the geometric means of the numbers from to for =, …,. Equality holds if the matrix is a block diagonal matrix consisting of matrices of ones or results from row and/or column permutations of such a block diagonal matrix.
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