Yifeng Huang, USC 

Title: A quadratic form interpolating \texttt{dinv}

Abstract:

The Catalan number C_n=\frac{1}{n+1}\binom{2n}{n} is a famous sequence that is the answer to numerous counting problems. Among them is the count of Dyck paths, namely, lattice paths above the diagonal in an n\times (n+1) rectangle. A celebrated refinement, called the q,t-Catalan number, counts Dyck paths while keeping track of two statistics: \texttt{area} and \texttt{dinv}. While \texttt{area} has a straightforward definition, \texttt{dinv} is somewhat curious and admits several (not obviously) equivalent definitions. Why \texttt{dinv} is important owes much to the symmetry of the q,t-Catalan numbers, a remarkable theorem of Garsia and Haiman. 

In this talk, we show that \texttt{dinv} has a new, "straightforward" definition using a quadratic form Q, which initially arises in my joint work on the geometry of singular curves with Jiang and Oblomkov. More generally, when evaluated on a pair of Dyck paths, the symmetric bilinear form is nonnegative and has a combinatorial interpretation we call "cross-\texttt{dinv}" (Theorem 1.2). We then use it to prove that Q is positive definite on a certain cone. 

The main engine, Theorem 1.2, is auto-formalized in Lean. This serves as an initial step in a bigger project with AxiomMath in the intersection of algebraic geometry and q-series.