A relationship between determinants and permanents

Mathematical Notes - A class of matrices is determined for which a Polya transformation of permanents into determinants can be defined.     

A note on a conjecture on permanents

Let Kn denote the set of all n × n nonnegative matrices whose entries have sum n, and let ϕ be a real function on K_n defined by ϕ (X) = Πⁿ_i=1Σⁿ_j=1x_ij + Πⁿ_j=1Σⁿ_i=1x_ij − per X for X = [x_ij] ϵ K_n. A matrix A ϵ K_n is called a ϕ -maximizing matrix on K_n if ϕ (A) ⩾ ϕ (X) for all X ϵ K_n. It is conjectured that J_n = [1/n]_{n × n} is the unique ϕ-maximizing matrix on K_n. In this note, the following are proved: (i) If A is a positive ϕ-maximizing matrix, then A = J_n. (ii) If A is a row stochastic ϕ-maximizing matrix, then A = J_n. (iii) Every row sum and every column sum of a ϕ-maximizing matrix lies between 1 − √2·n!/nⁿ and 1 + (n − 1)√2·n!/nⁿ. (iv) For any p.s.d. symmetric A ϵ K_n, ϕ (A) ⩽ 2 − n!/nⁿ with equality iff A = J_n. (v) ϕ attains a strict local maximum on K_n at J_n.     

A stronger form of the Egorychev-Falikman theorem on permanents

If A is a doubly stochastic matrix, it is shown that under certain conditions, there exist i, j such that the matrix obtained by replacing both the ith column of A with their average has a smaller permanent than that of A. A result which is stronger than the Egorychev-Falikman theorem on permanents is also proved.     

An identity involving permanents

An identity is stated which gives an expansion in terms of permanents for the reciprocal of the characteristic polynomial of a square matrix.     

Bounds for permanents and determinants

An inequality of Johnson and Newman for determinants of real matrices is extended to complex matrices. A related inequality for permanents of real matrices is improved by means of a new rearrangement theorem.     

Random permanents of mixed sample matrices

We study the central limit theorem for mixed random permanents.     

Normal approximation of random permanents

For random permanents, we obtain an estimate of the rate of convergence in the central limit theorem.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 922–927, July, 1995.     

A note on the analogue of Oppenheim's inequality for permanents

The inequality per(A∘B)⩽ perA perB is verified for nonnegative 2×2 and 3×3 Hermitian matrices, where A∘B is the Hadamard product ofA and B.     

On permanents of positive semidefinite matrices

It is shown in an elementary way that if A and B are positive semidefinite matrices, then per(A + B) ? per A + per B. The conditions under which equality may occur in this inequality are completely described, and some consequences are given.     

Expressions for certain minors and permanents

We provide explicit formulas for the minors of I–A and for per(I–A) where A is a row stochastic matrix.     

Multidimensional permanents in enumeration problems

Permanents are an effective tool for solving many combinatorial problems about enumeration. The respective theory is well developed and has numerous applications. In this paper the problem of counting the number of distinct 1-perfect binary codes is reduced to computing a generalized permanent of a particularly constructed multidimensional matrix.     

Inequalities for determinants and permanents

Let p be a prime > 3. It is shown that no integral circulant of order p^k exists with determinant p^k+1 . It is also shown that m is the determinant of an integral 9×9 circulant if and only if (m, 3)=l, or m = 0 mod 27. The proof makes use of a criterion which must be satisfied by the difference of two units in the cyclotomic field of level p^k .     

Random permanents of mixed multisampling matrices

We consider the weak convergence of random permanents under certain conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 7, pp. 1002–1005, July, 1995.     

Random permanents and symmetric statistics

Permanents of matrices with random elements are studied. For random permanents of finitedimensional projection matrices, new limiting results are obtained in different asymptotic schemes. The principal statistical functionals such asU-statistics, symmetric statistics, and others, are represented as functionals of permanent measures.Part of this research was supported by the Fund for Fundamental Researches of Ukrainian State Committee of Science and Technology.     

Theory of permanents 1982-1985

This is a survey of developments in the theory of permanents during the last quadrennium. It is a sequel to the monograph Permanents and the survey article Theory of Permanents 1978-1981. Many of the significant results obtained during the last four years are described in some detail. The current status of the conjectures and unsolved problems listed in previous surveys is described, and new conjectures proposed during the present period are listed. A comprehensive bibliography is appended. together with a list of addenda to previous bibliographies. Each item on either list is followed by an abstract.