On immanants of Jacobi-Trudi matrices and permutations with restricted position

Abstract

Let [chi] be a character of the symmetric group Ln. The immanant of an n x n matrix A = [aij] with respect to [chi] is [Sigma]w [epsilon] Sn [chi](w) a1,w(1) ... an,w(n). Goulden and Jackson conjectured, and Greene recently proved, that immanants of Jacobi-Trudi matrices are polynomials with nonnegative integer coefficients. This led one of us (Stembridge) to formulate a series of conjectures involving immanants, some of which amount to stronger versions of the original Goulden-Jackson conjecture. In this paper, we prove some special cases of one of the stronger conjectures. One of the special cases we prove develops from a generalization of the theory of permutations with restricted position which takes into account the cycle structure of the permutations. We also present two refinements of the immanant conjectures, as well as a related conjecture on the number of ways to partition a partially ordered set into chains.