Towards a q-analogue of the Harer-Zagier formula via rook placements

In 1986 Harer and Zagier computed a certain matrix integral to determine an influential closed-form formula for the number of (orientable) one-face maps on n vertices colored from N colors. Kerov (1997) provided a proof which computed the same matrix integral differently, which gave an interpretation of these numbers as also counting the number of placements of non-attacking rooks on Young diagrams. Bernardi (2010) provided a bijective proof of this formula by putting one-face maps in bijection with tree-rooted maps, which are orientable maps with a designated spanning tree. In the first part of the paper, we explore the connection between these rook placements and tree-rooted maps by developing a bijection between these objects. Rook placements on Young diagrams have a q-analogue due to Garsia and Remmel (1986). In the second part of the paper, we propose a statistic on rook placements that leads to a conjectured identity which is a q-analogue of part of the Harer-Zagier formula. This identity is also expressed in terms of moments of orthogonal polynomials which are rescaling of q-Hermite polynomials. We then use these moments to give a recurrence for the proposed q-analogue.