math.CO — Combinatorics

The Macdonald $(q,t)$-permanent I: Algebra and positivity

Contributed by Suvrit Sra

We introduce the Macdonald $(q,t)$-permanent, a two-parameter cycle-weighted permanent that contains the usual permanent and determinant, as well as the $\alpha$-permanent of Vere-Jones, as degenerate cases. We prove a MacMahon-type master theorem for its generating function, and determine when it induces a valid kernel on tuples of vectors. Nonnegativity of the $(q,t)$-permanent itself on semidefinite matrices turns out to be subtler: for $t=q^\alpha$ nonnegativity holds exactly at the positive Wallach exponents; for a fixed $q$ we determine the region in rank two and for higher ranks under an additional condition, while for small $q$ positivity can even reach beyond the kernel region. Whether positivity forces $t\in\{0,q,q^2,\ldots\}$ remains an intriguing open problem.