As Above, So Below: (m)-associahedra and (m)-noncrossing partitions polynomials

Finally, I’ve begun to morph my notes in raw LaTex from my early March posts on this topic into pdf files. The notes have become rather long, so I’m breaking them up into several p.d.f.s, each containing some coherent section of the previous notes with editing and some updates.

For any integer, this first section characterizes these partition polynomials (ParPs), multivariate in an infinite set of commutative indeterminates, via associated algebraic and differential identities. Later sections will contain reductions of these multivariate ParPs into single variable polynomials, relations among the diagonal coefficients and sums of the coefficients of the ParPs, and explicit lists of the first few of the ParPs and their reductions. References to related literature will be given.

As Above, So Below: (m)-associahedra and (m)-noncrossing partitions polynomials, Section 1 (a pdf)

In the appendix

A Schur Thing – Appendix to As Above, So Below: The Up-Down Operators for the (m)-Associahedra Partition Polynomials (pdf)

a Lagrange-Schur-Jabotinsky identity relating different coefficients of a power series raised to different integer powers is used to prove the raising and lowering operations of the sets of (m)-associahedra partition polynomials are related to the set of noncrossing partition polynomials (both sets introduced in various previous posts) by

and

Consequently, the sets of (m)-noncrossing partition polynomials, which satisfy the identities

,

also satisfy

for any integer.

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Related MathOverflow questions and answers:

“Interpretations of the (−m)-Fuss-Narayana numbers for m>0“, one of mine (April 2023), removed from MO-Q and ported to my WordPress mini-arXiv.

“Cataland: Facets and partition polynomials of cluster complexes“, one of mine (April 2023)

“Intersection numbers of moduli spaces and noncrossing partitions“, one of mine (April 2023)

“Log associahedra and log noncrossing partitions–raising ops and symmetric function theory for An (references)“, one of mine (March 2023)

“A combinatorial interpretation for n-ary trees for negative n” MO-Q posted by Alexander Burstein, to which I contributed (March 2023).

“A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)“, one of mine (March 2023)

“Combinatorics of iterated composition of noncrossing partition polynomials“, one of mine (June 2022)

“Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution“, one of mine May 2022)

“Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory“, one of mine (Dec 2021)

“Guises of the noncrossing partitions (NCPs)“, one of mine (Aug. 2019)

“Refined f- and h-partition polynomials of the associahedra“, one of mine (June 2018)

“Guises of the Stasheff polytopes, associahedra for the Coxeter An root system?“, one of mine (Oct. 2014)

“Why is there a connection between enumerative geometry and nonlinear waves?“, posted by
Nathaniel Bottman (Sept. 2014)

“Combinatorics of the Stasheff polytopes“, posted by Somnath Basu (Oct. 2013)

“What is Lagrange inversion good for?” posted by David Speyer (2010), my answers (2011, March 2023)

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Some of my related WordPress posts:

Every post since “A Doubly Infinite Ladder” (Mar. 2023)

“The Gang of Five: A series inversion group” (Aug. 2022)

“Schur expansion coefficients, generalized Faber polynomials, convolutions, and umbral identities” (Aug. 2022)

“Generalized Schur expansion coefficients” (Aug. 2022)

“One Matrix to Rule Them All” (July 2022)

“Matryoshka Dolls: Iterated noncrossing partitions, the refined Narayana group, and quantum fields” (July, 2022)

“A Taste of Moonshine in Free Moments” (Jan. 2022)

“A Doubly Infinite Ladder: The m-associahedra, the m- noncrossing partitions, raising and lowering operations, and combinatorial reciprocity for m an integer“