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