Abstract:
The Hafnian was introduced in the middle of 20th century by the Italian theoretical physicist E. R. Caianiello in connection with problems of quantum field theory. Later, the Hafnian found its application in combinatorics as an enumeration function of the number of perfect matchings of graphs. In its form, the Hafnian is close to such a better-known function as the Pfaffian. Unlike the latter, it is defined on symmetric, not skew-symmetric matrices and does not take into account the signs of permutations of indices in the corresponding monomials. In this paper, we consider the $q$-Hafnian, a generalization of the Hafnian that depends on formal parameters and coincides with the original function for unit values of the parameters. We indicate the combinatorial meaning of the $q$-Hafnian as a generating function of the number of permutations and the number of arc (linear chord) diagrams of certain classes. We prove several properties of the one- and two-parameter $q$-Hafnian that are generalizations of the properties of the usual Hafnian. In particular, we present an analog of the row decomposition property and an analog of the property expressing the Hafnian of the adjacency matrix of a bipartite weighted graph with equal parts through the permanent of the biadjacency matrix. These concepts and properties, in addition to their purely theoretical interest, can be used in developing algorithms that study the statistics of the number of inversions of certain classes of permutations and the statistics of the number of crossings and nestings in various classes of arc diagrams.
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