Гафниан двухпараметрических матриц

Abstract

The concept of the hafnian first appeared in the works on
 quantum field theory by E. R. Caianiello. However, it also has
 an important combinatorial property: the hafnian of the adjacency
 matrix of an undirected weighted graph is equal to the
 total sum of the weights of perfect matchings in this graph.
 In general, the use of the hafnian is limited by the complexity
 of its computation. In this paper, we present a method for the
 exact calculation of the hafnian of two-parameter matrices.
 In terms of graphs, we count the total sum of the weights of
 perfect matchings in graphs whose edge weights take only
 two values. This method is based on the formula expressing
 the hafnian of a sum of two matrices through the product of
 the hafnians of their submatrices. The necessary condition
 for the application of this method is the possibility to count
 the number of k-edge matchings in some graphs. We consider
 a special case in detail using a Toeplitz matrix as the
 two-parameter matrix. As an example, we propose a new interpretation
 of some of the sequences from the On-Line Encyclopedia
 of Integer Sequences and then provide new analytical
 formulas to count the number of certain linear chord
 diagrams.