Related papers: Some algebraic identities for the alpha-permanent
We explore some properties of a recent representation of permanental vectors which expresses them as sums of independent vectors with components that are independent gamma random variables.
In this paper, it is shown that $\alpha$-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of $\alpha$-permanents of the block matrices obtained from matrices associated to $*$-forests, which…
There is a digraph corresponding to every square matrix over $\mathbb{C}$. We generate a recurrence relation using the Laplace expansion to calculate the characteristic, and permanent polynomials of a square matrix. Solving this recurrence…
We prove an identity relating the permanent of a rank $2$ matrix and the determinants of its Hadamard powers. When viewed in the right way, the resulting formula looks strikingly similar to an identity of Carlitz and Levine, suggesting the…
In this note we settle two open problems in the theory of permanents by using recent results from other areas of mathematics. Bapat conjectured that certain quotients of permanents, which generalize symmetric function means, are concave. We…
In this paper, we give some determinantal and permanental representations of Generalized Lucas Polynomials by using various Hessenberg matrices, which are general form of determinantal and permanental representations of ordinary Lucas and…
We develop an abstract look at linear optical networks from the viewpoint of combinatorics and permanents. In particular we show that calculation of matrix elements of unitarily transformed photonic multi-mode states is intimately linked to…
We recall Vere-Jones's definition of the $\alpha$--permanent and describe the connection between the (1/2)--permanent and the hafnian. We establish expansion formulae for the $\alpha$--permanent in terms of partitions of the index set, and…
The alpha-determinant unifies and interpolates the notion of the determinant and permanent. We determine the irreducible decomposition of the cyclic module of $gl_n(C)$ defined by the alpha-determinant. The degeneracy of the irreducible…
The fermionant can be seen as a generalization of both the permanent (for $k=-1$) and the determinant. We demonstrate that it is VNP-complete for most cases. Furthermore it is #P-complete for the cases. The immanant is also a generalization…
We describe immanants as trivial modules of the symmetric group and show that any homogeneous polynomial of degree n on the space of n by n matrices preserved up to scalar by left and right action by diagonal matrices and conjugation by…
In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite…
In this work we recall the definition of matrix immanants, a generalization of the determinant and permanent of a matrix. We use them to generalize families of symmetric and antisymmetric orbit functions related to Weyl groups of the simple…
We present identities for permutations with fixed points. The formulas are based on successive derivations or integrations of the determinant of a particular matrix.
Immanants are polynomial functions of n by n matrices attached to irreducible characters of the symmetric group S_n, or equivalently to Young diagrams of size n. Immanants include determinants and permanents as extreme cases. Valiant proved…
We show that the permanent of a matrix can be written as the expectation value of a function of random variables each with zero mean and unit variance. This result is used to show that Glynn's theorem and a simplified MacMahon theorem…
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…
The paper addresses the calculation of correlation functions of permanental polynomials of matrices with random entries. By exploiting a convenient contour integral representation of the matrix permanent some explicit results are provided…
Let $X$ be a matrix of indeterminates, $t$ an integer, and $P_t(X)$ define the ideal generated by the permanents of all $t\times t$ submatrix of $X$. $P_t(X)$ is called a permanental ideal. In this article, we study the algebras…
Let $R$ be a commutative additively idempotent semiring. In this paper, some properties and characterizations for permanents of matrices over $R$ are established, and several inequalities for permanents are given. Also, the adjiont matrices…