Resume for the (m)-associahedra and (m)-noncrossing partitions polynomials

Raw LaTex pdf, to be published later today and extended over the next couple of weeks. Includes several analytic identities and compilations of the first few -associahedra and -noncrossing / Narayana partition polynomials from to .

Resume for the -associahedra and -noncrossing partitions polynomials (soon)

(I don’t think I’ve suffered any drain bamage recently, but crossing the i’s and dotting the t’s is taking some mite–and I keep discovering new intriguing paths to explore. )

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Laurent series and o.g.f.s and e.g.f.s, oh my! Narayana and Catalan versus Euler and Grassmann

Excitedly skipping down the yellow-brick road, I’m making another quick post in raw LaTex. This time on the parallelism between the dual noncrossing partitions and associahedra for compositional inversion of o.g.f.s (and Laurent series) and the dual refined Eulerians and classic Lagrange inversion polynomials for compositional inversion of e.g.f.s (and Laurent series). Although the indeterminates for the partition polynomials of one dual can be changed into those of the other by a simple scaling with the factorials (or a formal Borel-Laplace transform), the beasts that issue forth are quite different in character.

The naturally reduced noncrossing partition polynomials (A134264), or refined Narayana polynomials, are the shifted Narayana polynomials (A001263), whose coefficients sum to the Catalan numbers (A000108). The naturally reduced associahedra polynomials (A133437 / A111785) become the signed, face polynomials of the associahedra (A033282 / A126216 ) whose vertices are enumerated by the Catalan numbers–the coefficients are related to dissections of convex polygons and trees, of course. The Lagrange inversion partition polynomials (A134685) actually have a simple combinatorial model of balls in a bin. Their natural reductions (the Ward polynomials of A134991) are associated with simplicial complexes, the tropical Grassmannians , and phylogenetic trees among other constructs. The refined Euler polynomials (A145271) reduce, naturally, to the Eulerian polynomials (A008292), which are the h-polynomials for the permutahedra (A019538) or their dual simplicial complexes, but the second-order Eulerian polynomials (A008517) are the h-polynomials for the tropical Grassmannians . (I’m being a little loose in the connections–in some cases the f-polynomials / f-vectors and h-polynomials / h-vectors need to have their coefficients in reverse order to comply with the standard literature on the topics, but this is simple change of variables.)

Laurent series and o.g.f.s and e.g.f.s, oh my! Narayana and Catalan versus Euler and Grassmann (a pdf)

Combinatorics for a generalized Lagrange inversion formula and the (±m)-associahedra and (±m)-noncrossing partitions

I just posted (Mar. 15, 2023) a MO-Q that is related to the my last two posts on this blog. The MO-Q relates the different methods of compositional inversion manifest in the different sets of partition polynomials to one overarching analytic expression generating the sets of m-associahedra and m-noncrossing partition polynomials for m any integer. This accentuates the connections of the rising and falling factorials in binomial expansions to a generic combinatorial-analytic reciprocity:

“Combinatorics for a generalized Lagrange inversion formula and the (±m)-associahedra and (±m)−noncrossing partitions“

March 17, 2023 (St. Patrick’s Day–as if the Irish ever needed an excuse to drink.): The background for the question is rather long, the question generated no interest on MO-Q, and I’m making progress on different models myself and extension to noncommutative symmetric functions, so I’ve removed the question from MO and pasted it into a pdf file in raw LaTex.

As Above, So Below: Dualities, combinatorial reciprocities, and the associahedra and noncrossing partitions of the Weyl-Coxeter group A_n

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).

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

Based on the inversion algebra I’ve presented in previous posts, I answered an inspiring question by Alexander Burstein on MathOverflow on March 1 on a specific combinatorial reciprocity. The question is really the tip of an iceberg and my answer and a more general associated question quickly emerged from the depths to become too large for MO to comfortably envelope, so I’ve tugged it over to the deeper waters here. I haven’t had time to convert the notes into proper format yet, but here are the notes in raw LaTex that can be copied and pasted into a Q&A window that can render LaTex .

These will soon be revised with better associations in signs and indices at different levels of refinement and representations (I hope–lost one revamp already).

An answer to the question by Alexander Burstein: The Iceberg

The associated question: Extension of a combinatorial reciprocity to the symmetric functions of Novelli and Thibon with noncommuting indeterminates

In light of the relation between the two sets of Fuss-Catalan sequences–the rising and the falling– to the relation between the rising and falling factorials, , I wouldn’t be surprised if the Lah polynomials can be worked into the picture (see one of my answers and also Sam Hopkins’ to the MO-Q “Important formulas in combinatorics” and OEIS A008297, A111596, A105278, and A130561. The Lah polynomials are related to the theory of conformal complex functions and via and $latex((1+x)^2D_x$ and are the base sequence for the associated Laguerre polynomials which generalize to confluent hypergeometric functions important in pretty much any physics involving vibrations / oscillatory phenomena.

Edit: March 13, 2023

For more thoughts on the rising and falling factorial duality. See the MO-Q “A bridge between the algebraic / differential geometry of sl2(C) and the Sheffer-Appell calculus and combinatorics“

Compilation of OEIS Partition Polynomials A133314, A134685, A145271, A356144, and A356145

The pdf below contains a compilation of the first eight polynomials of each of the five sets of partition polynomials presented in the OEIS entries listed above, related to compositional and multiplicative inversion of formal Taylor series, or exponential generating functions. The sets are depicted twice–first in standard math format and then in raw LaTex.

Compilation of OEIS Partition Polynomials A133314, A134685, A145271, A356144, and A356145

 March 4, 2023: Upon reading a question I posted on MathOverflow on generalization of the face-h polynomial identity for simplicial complexes, a generalization based on the inversion group that these partition polynomials encompass, Richard Stanley made use of the set of Lagrange inversion polynomials in testing a conjecture he had on the positivity of under transformation of the indeterminates into power sums and their ultimate reduction to elementary symmetric functions. In the process he and I discovered some typos in the monomials of , which I have corrected. Upon correction, his conjecture on e-positivity turns out to be true up through , making it almost a sure bet that it’s true in general.

The Bernoulli function: Integration, summation, and the Hurwitz and Riemann zeta functions

These are some notes I made at the end of 2020 that I’ve meant to return to and brush up. Need further work. (Another argument for the relation between the Riemann sum and the Bernoulli function for the integral of with complex added on 9/30/2022.)

The Bernoulli function: Integration, summation, and the Hurwitz and Riemann zeta functions

Related MO-Qs:

Intuitive explanation why “shadow operator” D/(e^D−1) connects logarithms with trigonometric functions?

Ramanujan’s Master Formula: A proof and relation to umbral calculus

The Norlund Self-Convolutional Bernoulli Polynomials and Derivatives as Finite Differences

Sets of Norlund polynomials are formed by iterated self-convolutions of the set of Bernoulli polynomials and can be used to express the derivatives of the powers in terms of finite differences and, therefore, formally, the derivatives of power series. As one illustration / check of this formalism, I give an expression for the Fuss-Catalan sequences in terms of finite differences of the Norlund sequences of polynomials evaluated at positive integers in the following pdf;

The Norlund Self-Convolutional Bernoulli Polynomials and Derivatives as Finite Differences.

(This pdf was originally written in 2020. I’ve corrected a typo and have extended the formalism for the Catalan numbers to the Fuss-Catalan numbers and, in the process, have had to change the formatting slightly since the functionality of my LaTex app has been reduced in the interim.)

The reduced inverse refined Eulerian polynomials and associated arrays

The pdf below goes into detail on a reduced form, or specialization, of the inverse refined Eulerian polynomials (OEIS A356145, in draft), introduced in my previous post “Matryoshka Dolls …”, which constitute the inverse of the refined Eulerian polynomials of A145271 with respect to indeterminate substitution.

The reduced inverse refined Eulerian polynomials and associated arrays (pretty much stable version finished Aug. 20, 2022)

Supplement (Aug. 20, 2022): The following pdf gives a differential equation defining a set of Appell Sheffer polynomials that provide a parametrized trajectory between the associated Bell polynomials of A008299 and the associated reduced inverse refined Eulerian polynomials of A124324, discussed in the notes above.

Appell-Bell polynomials: Linking the associated Bell polynomials and the associated reduced inverse refined Eulerian polynomials (pdf) (there are some obvious typos in this pdf)

The Gang of Five: A series inversion group

The gang of five is an inversion group of partition polynomials intimately interlinked via three pairs of compositionally inverse series–two pairs of Laurent series and one of power series, or o.g.f.s–and the series expansion of reciprocals. The group consists of the associahedra, refined Narayana (noncrossing partition), refined inverse Narayana, special Schur expansion coefficient, and reciprocal polynomials.

“The Gang of Five: Inversion Group of Partition Polynomials and Three Pairs of Compositionally Inverse Series” (pdf)_