Partition statistics counting the Betti numbers of equivariant Hilbert schemes

In this thesis, we study two collections of partition statistics (one collection labelled with a +, the other with a -) parameterised by a positive real number x, a positive integer c, which count the Betti numbers of Hilbert schemes of points on a surface, equivariant under the action of a cyclic group, giving a bijective proof that the statistics are equidistributed over partitions of n with a fixed c-core.

We extend techniques introduced by Loehr and Warrington to work with cores and quotients of partitions, and define a map from partitions to multigraphs (where the partition is viewed as an Eulerian tour of the graph) that retains the relevant data on partitions, including
the c-core, size, and other statistics closely related to the family of statistics of interest. We then define a bijection on partitions with a fixed c-core that preserves the multigraph, which we use to prove bijectively that the distribution of our statistics is independent of x and the sign. We also compute the x = 0 case to obtain a generating function for the distribution.

As a byproduct, we give a combinatorial proof of a partition identity of Buryak-Feigin-Nakajima that can be seen as giving the Betti numbers of C∗-equivariant Hilbert schemes.