(m)-Associahedra and (-m)-Noncrossing Partition Polynomials as Moments of Umbral Inverse Appell Polynomial Sequences

Numerous relations between the pair of sets of (m)-associahedra and  (-m)-noncrossing partitions / (-m)-Narayana partition polynomials can be deduced from treating them as the moments of an umbral inverse pair of Appell-Sheffer polynomials sequences.

(m)-Associahedra and (-m)-Noncrossing Partition Polynomials as Moments of Umbral Inverse Appell Polynomial Sequences (a pdf)

The Appell-Sheffer polynomial calculus is intimately related to that of symmetric function theory, as I’ve demonstrated in numerous posts. The Appell raising operators provide the recursion relations interweaving the formal power sums, or Faber polynomials, and the elementary symmetric polynomials / functions and interweaving the power sums and the complete homogeneous polynomials / functions mod sign conventions and normalization factors. Compare the brief presentation of the Appell formalism in the pdf with the Wikipedia article on the Newton identities. The action of the raising op for an Appell sequence and that for its umbral inverse Appell sequence evaluated at give the recursion relations, again mod signs and factorial normalization. OEIS A263916 has more detail and references on the Faber polynomials, defined in terms of the logarithmic derivative of an o.g.f. (compare with A263634).

Supplement (added My 7, 2023):

Some identities relating the Appell polynomial formalism to the power sums / Faber polynomials and elementary symmetric polynomials / functions are reprised in

Appell-Sheffer and Symmetric Polynomials (pdf).

The exponential class of compositional partition polynomials presented in this pdf,

1. Stirling partition polynomials of the first kind, a.k.a. the cycle index polynomials of the symmetric groups of A036039
2. Stirling partition polynomials of the second kind, a.k.a. the Faa di Bruno / Bell partition polynomials of A036040
3. Lah partition polynomials of A130561,

can be cast as either binomial Sheffer sequences in a single variable or as Appell Sheffer sequences in a distinguished indeterminate.

The indeterminates of the three sets of compositional partition polynomials of the exponential family can be extracted using the two sets of compositional partition polynomials of the logarithmic family

1) the cumulant expansion partition polynomials of A127671, a.k.a. the logarithmic polynomials of A263634

2) the Faber partition polynomials of A263916,

and vice versa.

Numerous diagrammatics are associated with the classical moment-cumulant relations related to the inverse pair the Bell / Faa di Bruno partition polynomials of A036040 and cumulant expansion partition polynomials of A127671 / logarithmic polynomials of A263634, which can be used to characterize the Appell raising op.

Darij Grinberg commented that the recursion relation for the elementary symmetric functions involving products of the powers sums and the lower order elementary sym. fcts. has a combinatorial proof in An Introduction to Symmetric Functions and Their Combinatorics, AMS 2019, by Egge (see eqns. 3.6 and 3.7 on p. 56).

The elementary, complete homogeneous, and power sums polynomials occur on pg. 52 (eqns. A.22-24) of “Classical -geometry” by Gervais and Matsuo with Schur polynomials defined on pg. 26 (eqn. 3.39, also see second paragraph on pg. 53 and OEIS A036039).