A matrix of permanents

If A is a matrix of order n×(n?2), n?3, denote by ā the n×n matrix whose (i,j)th entry is zero if i=j, and if i≠j, is the permanent of the submatrix of A obtained by deleting its ith and jth rows. It is shown that if A is a nonnegative n×(n?2) matrix, then ā is nonsingular if and only if A has no zero submatrix of n?1 lines. This is used to give precise consequences of the occurrence of equality in Alexandroff's inequality.     

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.     

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.     

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.     

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.     

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.     

Variation of symmetric tensor powers and permanents

The norm of the derivative of the symmetric tensor power of an operator is evaluated exactly. A bound for the distance between the permanents of two matrices is obtained as a consequence.