We prove that for writing the 3 by 3 permanent polynomial as a determinant of a matrix consisting only of zeros, ones, and variables as entries, a 7 by 7 matrix is required. Our proof is computer based and uses the enumeration of bipartite graphs. Furthermore, we analyze sequences of polynomials that are determinants of polynomially sized matrices consisting only of zeros, ones, and variables. ...

In 1992, Goulden and Jackson found a beautiful determinantal expression for the immanant of a matrix. This paper proves the same result combinatori-ally.

This work is motivated by the relation between KP and BKP integrable hierarchies, whose τ-functions may be viewed as flows of sections dual determinantal Pfaffian line bundles over infinite dimensional Grassmannians. In finite dimensions, we show how to relate Cartan map, which, for a vector space V dimension N, embeds Grassmannian GrV0(V+V*) maximal isotropic subspaces + V*, with respect natur...

We prove that the correlations functions, generated by the determinantal process of the Christoffel-Darboux kernel of an arbitrary order 2 ODE, do satisfy loop equations.