Haglund–Haiman–Loehr diagonal-coinvariant Hilbert-series conjecture

About 12 years old · traced to

Let PFn\mathsf{PF}_n be the set of labeled classical parking functions. For P∈PFnP\in\mathsf{PF}_n, let area(P)\mathsf{area}(P) be its area statistic and let dinv(P)\mathsf{dinv}(P) be its diagonal inversion statistic. Let DRn\mathsf{DR}_n be the doubly graded diagonal coinvariant ring. Haglund–Haiman–Loehr Hilbert-series conjecture. For all n>0n>0,

HilbDRn(q,t)=∑P∈PFnqarea(P)tdinv(P).\mathsf{Hilb}_{\mathsf{DR}_n}(q,t)=\sum_{P\in\mathsf{PF}_n}q^{\mathsf{area}(P)}t^{\mathsf{dinv}(P)}.

This conjecture gives a combinatorial formula for the doubly graded Hilbert series of diagonal coinvariants; the supplied text does not state its resolution.

References

Primary source

Drew Armstrong, Nicholas A. Loehr and Gregory S. Warrington, “Rational parking functions and Catalan numbers”, arXiv:1403.1845 (2014).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.