Rook placements and Jordan forms of upper-triangular nilpotent matrices

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. We present a combinatorial formula for computing the number F_\lambda(q) of matrices of Jordan type lambda as a weighted sum over standard Young tableaux. We also study a connection between these matrices and non-attacking rook placements, which leads to a refinement of the formula for F_\lambda(q).Comment: 25 pages, 6 figure

Fermions in Deep Optical Lattice under p-wave Feshbach Resonance

We investigate theoretically the energy levels of two identical Fermions in a harmonic potential well under p-wave Feshbach resonance. The magnetic fields needed to affect the energy levels differ strongly among the levels, and they can be at values far from the free space resonant field.Comment: 2 figure