Mahir Bilen Can

Nested Hilbert Schemes and the nested $q,t$Catalan series
dmtcs:3636 
Discrete Mathematics & Theoretical Computer Science,
January 1, 2008,
DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)

https://doi.org/10.46298/dmtcs.3636
Nested Hilbert Schemes and the nested $q,t$Catalan seriesArticle
In this paper we study the tangent spaces of the smooth nested Hilbert scheme $\mathrm{Hilb}^{n,n1}(\mathbb{A}^2)$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\mathbb{T}^2$equivariant locally free sheaf on $\mathrm{Hilb}^{n,n1}(\mathbb{A}^2)$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with nonnegative integer coefficients. We call this conjecturally positive polynomial as the "nested $q,t$Catalan series,'' for it has many conjectural properties similar to that of the $q,t$Catalan series.