The Scottish Cafe

All but two or three of my posts have been on mathematics that I have explored. This one is a deviation made two years after the invasion of Ukraine by the shameless autocrat Putin and at a time when one of his shameless admirers is seeking the U.S. Presidency again. Democracy, which invigorates creativity and is invigorated by it, a beautiful synergy, is endangered globally yet again. Perhaps it’s worth the time to reflect on a time and place of freedom and creativity between the World Wars that was destroyed by the rise of two dictatorships with the hope that history doesn’t repeat itself yet again, not here at least.

Mathematicians at the Scottish Café by Chris Zielinski

BEAUTIFUL MINDS. EXTRAORDINARY MATHEMATICIANS FROM LWÓW, an interview with Mariusz Urbanek

The Lwów School of Mathematics. a post by Mariusz Urbanek

2014, the year of the Maidan Revolution, the Revolution of Dignity, birthing democracy with courage, blood, and tears of grief and joy in Ukraine is also the year the Scottish Cafe was re-established in Lviv, Ukraine, formerly Lwow of Poland.

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

This is a pointer to the appendix

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

to the earlier set of notes “As Above, So Below . . .”. (A link to this pdf is provided in that post as well.)

In these notes, 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.

(m)-Associahedra and (m)-Noncrossing Partition Polynomials and the Infinite Dihedral Group

I was playing around once again yesterday with the basic algebraic relations among the sets of (m)-associahedra partition polynomials and the sets of (m)-noncrossing partitions polynomials , which I’ve presented in several posts over the last year or so, and decided to post a few defining relations for the dual sets of polynomials in a question on Math Stack Exchange in the hope that the group would be recognized by someone. The MSE user Karl pointed out that it sounds like I was describing the infinite dihedral group, linking to the associated Wikipedia article. This led me to the Wiki on the dihedral group and a set of group relations that are shared by my group of partition polynomials. Each set plays the role of a rotation in the dihedral group and each set , a reflection . In the following I’ll show the applicability of these relations under the substitution operation I’ve illustrated in previous posts.

An infinite group is formed by iterating the substitution operation on and its inverse . The elements of this infinite group are where is any integer; , the identity; and, e.g., and under the repeated operation.

The infinite sets of (m)-associahedra partition polynomials satisfy for any integer

(I)

and

(II)

That is, is involutive and and are the ladder ops–the raising and lowering ops–for the infinite set comprised of the infinite sets , where runs over the infinite set of integers, as well as the ladder ops for the group .

For any integers and , clearly, by the definition of above,

and the relation

implies

or equivalently

Recalling , so , and taking the inverse of this last equality gives

implying

Similarly, since

finally

Consequently, with and , my group satisfies the four relations

presented in the Wikipedia article on the dihedral group.

The group also satisfies the conjugation relations

1)

and

2)

An analogous algebraic realization of the infinite dihedral group can be obtained by scaling the indeterminates by starting with multiplicative and compositional inversion of exponential generating functions (Taylor series) rather than ordinary generating function (power series). The extrapolation in both cases includes Laurent series. It would be interesting if there were other multivariate realizations under substitution.

Related stuff:

“Noncrossing partitions under rotation and reflection” by Callan and Smiley (pg.6)

“The power of group generators and relations: an examination of the concept and its applications” by Zhou

The Geometry and Topology of the Coxeter Groups by Davis

Group Theory by Milne

The CRM Winter School on Coxeter groups

Generalized Dihedral Group at GroupProps

“Dihedral group” by K. Conrad

Iterative formula for the Kreweras-Voiculescu polynomials–the noncrossing partitions polynomials

The Kreweras-Voiculescu partition polynomials, which I usually call the noncrossing partitions or the refined Narayana partition polynomials of OEIS A134264–flag and enumerate distinct noncrossing partitions and give the free moments in terms of the free cumulants in free probability theory. They are often useful in mathematical / physical analyses involving compositional inversion and are also of broad use in combinatorics characterizing parking functions, trees, Dyck lattice paths, cluster complexes, and other combinatorial geometric constructs. I’ve derived yet another method for iteratively determining these polynomials in the pdf

Iterative formula for the Kreweras-Voiculescu polynomials–the noncrossing partitions polynomials

Noncrossing partitions and (-1)-associahedra partition polynomials as Appell and quasi-Appell sequences

The following pdf concerns the celebrated partition polynomials of OEIS A134264 for compositional inversion of power series / ordinary generating functions (core partition polynomials in free probability theory, related to the combinatorics of Dyck paths, noncrossing partitions, parking functions, and other geometric constructs) and the (-1)-associahedra partition polynomials, which I have called the special Schur self-Konvolution partition polynomials in previous posts, for compositional inversion of a certain family of Laurent series;

Noncrossing partitions and (-1)-associahedra partition polynomials as Appell and quasi-Appell sequences

The umbral compositional inverse of a Sheffer polynomial sequence and its lowering and raising operators

The pdf

The umbral compositional inverse of a Sheffer polynomial sequence and its lowering and raising operators

reprises results of older notes.

See the MO-Q “Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)” for possible broader perspectives.

Reps of the lowering / destruction / annihilation operator for binomial Sheffer polynomial sequences

Just posted a belated response concerning these reps to the question “Trying to characterize an ;umbral shift‘” on Math Stack Exchange posted by the creator of the YouTube video The Shadowy World of Umbral Calculus. The analysis includes a presentation of the lowering op for binomial Sheffer sequences as a conjugation of the derivative op with the umbral substitution diff op for the sequence and a review of a derivation of the Bernoulli numbers from the Stirling numbers of the first and second kinds and the reciprocal integers.

A Catalan Identity Related to Fibonacci Polynomials and Its Generalization to the Fuss-Catalan Number Sequences

A simple identity for the classic Fuss-Catalan number sequences is addressed in

A Catalan Identity Related to Fibonacci Polynomials and Its Generalization to the Fuss-Catalan Number Sequences (a pdf)

Added June 25, 2023, an apology: I typically use only freely available apps to produce pdfs with LaTex conversion. I just noticed that the app I’m currently using with Google docs to convert the LaTex places a link / pointer to an online editor on every converted expression. Very annoying! (Other more user-friendly and efficient apps I’ve used in the past have developed paywalls or simply vanished…sigh.)

Lagrange à la Laguerre: Compositional inverse of a reduction of the (m)-noncrossing partitions / refined (m)-Narayana partition polynomials

Related to earlier posts on the (m)-associahedra and (m)-noncrossing partitions polynomials for compositional inversion of power and Laurent series. Provides a derivation of the compositional inverse of the power series of reduced (m)-noncrossing partitions polynomials via the classic operational calculus associated with the Lah polynomials–a variant of the associated Laguerre polynomials of order -1 and 1.

Lagrange à la Laguerre: Compositional inverse of a reduction of the (m)-noncrossing partitions / refined (m)-Narayana partition polynomials (a pdf, some obvious typos fixed June 14, 2023)

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)