Rational Parking Functions and LLT Polynomials

We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial. Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov. The corresponding skew diagram is described explicitly in terms of a certain (m,n)-core.Comment: 14 pages, 8 figure

Compactified Jacobians and q,t-Catalan Numbers, I

J. Piontkowski described the homology of the Jacobi factor of a plane curve singularity with one Puiseux pair. We discuss the combinatorial structure of his answer, in particular, relate it to the bigraded deformation of Catalan numbers introduced by A. Garsia and M. Haiman.Comment: Revised version, 24 page

Rational Dyck Paths in the Non Relatively Prime Case

We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb Z,$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$--Dyck paths and $d$-tuples of $(n,m)$-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.Comment: 25 pages, 9 figure

Capacitive coupling of atomic systems to mesoscopic conductors

We describe a technique that enables a strong, coherent coupling between isolated neutral atoms and mesoscopic conductors. The coupling is achieved by exciting atoms trapped above the surface of a superconducting transmission line into Rydberg states with large electric dipole moments, that induce voltage fluctuations in the transmission line. Using a mechanism analogous to cavity quantum electrodynamics an atomic state can be transferred to a long-lived mode of the fluctuating voltage, atoms separated by millimeters can be entangled, or the quantum state of a solid state device can be mapped onto atomic or photonic states.Comment: 4 pages, including one figure. v2: Improved discussion of surface effect