This is a draft of notes on the refined m-associahedra and m-Narayana / m-noncrossing partition polynomials (multivariate in an infinite number of independent, mutually commuting indeterminates), for any integer, and an underlying duality / combinatorial reciprocity reflected in their associated group algebra under substitution. Reduced versions or subsets of reduced versions of these polynomials in a single variable mostly for positive have been presented and discussed in one way or another by Drew Armstrong in his Ph.D. thesis, by Jean-Christophe Novelli and Jean-Yves Thibon, by Paul Barry, and by the collaborators Christos Athanasiadis, Henri Mühle, and Eleni Tzanaki.

This is supplemental to my last post “A Doubly Infinite Ladder” and again is in raw Latex, so it needs to be copied and pasted into a window that can covert it to normal math expressions, say, a Q & A window. Soon I hope to generate an Addendum with example computations and arrays to enhance confidence in the results (perhaps I’ll need to tweak the formulas after that) and that others can use to check any related proofs or conjectures they may come up with.

: Dualities, combinatorial reciprocities, and the associahedra and noncrossing partitions of the Weyl-Coxeter group (for pdf click on first few words).