A direct bijection for the Harer–Zagier formula

We give a combinatorial proof of Harer and Zagierʹs 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 B p , k , the enumeration of which is equivalent to the Harer–Zagier formula. The elements of B p , k are of the form ( μ , π ) , where μ is a pairing on { 1 , … , 2 p } , π is a partition into k blocks of the same set, and a certain relation holds between μ and π . (The set partitions π that can appear in B p , k are called “shift-symmetric”, for reasons that are explained in the paper.) The direct bijection for B p , k identifies it with a set of objects of the form ( ρ , t ) , where ρ is a pairing on a 2 ( p - k + 1 ) -subset of { 1 , … , 2 p } (a “partial pairing”), and t is an ordered tree with k vertices. If we specialize to the extreme case when p = k - 1 , then ρ is empty, and our bijection reduces to a well-known tree bijection.