On a generalisation of the Laguerre polynomials

Foata and Strehl found a beautiful combinatorial interpretation of the Laguerre polynomials: it turns out that these polynomials count the so-called Laguerre digraphs generalising digraphs of cyclic permutations. During my talk, I would like to introduce a multivariate generalisation of the Laguerre polynomials related to such graphs, as well as certain properties of matrices built from sequences of such polynomials. In particular, I will state conditions implying total nonnegativity (i.e. nonnegativity of all minors) of such matrices.