A direct bijection for the Harer-Zagier formula

We give a combinatorial proof of Harer and Zagier’s formula for the disjoint cycle distribution of a long cycle multiplied by an involution with no fixed points, in the symmetric group on a set of even cardinality. The main result of this paper is a direct bijection of a set Bp,k , the enumeration of which is equivalent to the Harer–Zagier formula. The elements of Bp,k are of the form ( , ), where is a pairing on {1, . . . , 2p}, is a partition into k blocks of the same set, and a certain relation holds between and . (The set partitions that can appear in Bp,k are called “shift-symmetric”, for reasons that are explained in the paper.) The direct bijection for Bp,k identifies it with a set of objects of the form ( , t), where is a pairing on a 2(p− k + 1)-subset of {1, . . . , 2p} (a “partial pairing”), and t is an ordered tree with k vertices. If we specialize to the extreme case when p = k − 1, then is empty, and our bijection reduces to a well-known tree bijection. © 2005 Elsevier Inc. All rights reserved. MSC: Primary, 05A15; Secondary, 05C05