Immanants of combinatorial matrices

Abstract

The arbitrary immanants of three matrices whose determinants are known to be generating functions for sets of combinatorial objects are examined. Combinatorial interpretations are given for the immanants of the Matrix-tree matrix, and a special case of the Jacobi-Trudi matrix. These allow us to deduce immediately the nonnegativity of the coefficients in the expansion of the immanants. A conjecture is made about the nonnegativity of coefficients of the expansion of the immanant of the Jacobi-Trudi matrix in the general case. This nonnegativity result is seen to fail for the Hankel matrix, and combinatorial reasons for this failure are given. All results can be translated into statements about the nonnegativity of Schur function expansions for the related symmetric functions.