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(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial formulas for the coefficients of the polynomials, extracted from the associated OEIS entries.)

In "Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra", Novelli and Thibon extend the set of noncrossing partition polynomials / refined Narayana polynomials $[N]$, multivariate in an infinite set of commuting, independent indeterminates to polynomials in the same indeterminates but with the restriction that they are noncommutative. $[N]$ is one set of Lagrange polynomials for compositional inversion of power series (and a family of Laurent series). A dual set for power series is that of the associahedra polynomials $[A]$. N & T can relate this to earlier work on $(m+1)$-Fuss-Catalan polynomials and $(m+1)$-Fuss-Narayana polynomials for $m > 1$--actually subsets of $[N]$ and $[A]$.

There is a conjugate dual, call it $[K]$, to $[A]$, a special set of self-Konvolution expansion coefficients among the infinite number of general sets developed from series expansions of $(1+ u_1 z +u_2 z^2 + \cdots)^m$ for $m$ any integer by Lagrange, Schur, and Jabotinsky. $[N]$ and $[A]$ are among these sets of expansion coefficients as well as $[N]^{-1}$, the inverse under substitution to $[N]$. The relations among $[N]$, $[A]$, $[N]^{-1}$, and $[K]$, can be derived either from their definitions as Lagrange-Schur-Jabotinsky expansion coefficients or from their definitions as compositional inversions of formal power or Laurent series.

The dual sets $[A^{(1)}]=[A]$ and $[N^{(1)}] =[N]$ can be extended very naturally to sets $[A^{(m)}]$ and $[N^{(m)}] = [N]^{m}$ for $m$ any integer where, in particular, $[A^{(-1)}]=[K]$ and $[N^{(-1)}] = [N]^{-1}$. Their natural reductions encompass all the mono-variable $(m+1)$-Fuss-Catalan and $(m+1)$-Fuss-Narayana polynomials and numbers discussed (at least subsets thereof) by N & T and in the additional papers listed below.

One duality for $m$ any integer, i.e., $m =0,\pm 1,\pm 2,...)$, is a generalized formal face-h-polynomial identity as a 'left-to-right reflection' between the duals $[A^{(m)}]$ and $[N^{(m)}]$, or, equivalently, as an up-down ladder identity for $[A^{m)}]$ with $[N]$ as the left-sided raising op and $[N]^{(-1)}$ as the left-sided lowering op:

$$[A^{(m)}] = [N]^m [A^{(0)}].$$

To some degree for $m >0$, this is reflected in the noncommutative symmetric analogues as Adams operations.

Question:

Has someone extended the noncommutative-indeterminates analogues of $[N^{(m)}]$ and $[A^{(m)}]$ for $m > 1$ to $m \leq 0$?

(N & T have explicit examples of the noncommutative analogues of $[A]$, $[A^{(2)}]$, $[N]$, $[N]^2$, and $[N]^{-1}$).

Refs (most in N & T):

  1. "On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements" by Athanasiadis and Tzanaki

  2. "Generalized noncrossing partitions and the combinatorics of Coxeter groups" by Armstrong

  3. "Refined Lattice Path Enumeration and Combinatorial Reciprocity" by Mühle and Tzanaki

  4. "Polygon dissections and some generalizations of cluster complexes" by Tzanaki

  5. "On the inversion of Riordan arrays" by Barry

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  • $\begingroup$ This concerns o.g.f.s. There is a parallel development for e.g.f.s. that involves the refined Eulerian polynomials $[E]$, counterpart to $[N]$, and the classic Lagrange inversion polynomials $[L]$, counterpart to $[A]$. The counterpart to $[A^{(0)}] =[R]$, the polynomials for reciprocals of o.g.f.s is $[P]$, the permutahedra polynomials for the reciprocal of e.g.f.s. This involves the $G(2,n)$ Grassmannians, phylogenetic trees, and their cousins. $\endgroup$
    – Tom Copeland
    Commented Mar 18, 2023 at 23:25
  • $\begingroup$ $[K]= [A^{(-1)}] = [A^{(0)}][A][A^{(0)}]$ and $ [N]^{-1} = [A^{(0)}][N][A^{(0)}]$. Since $[A^{(m)}]$ is involutive under self-substitution, i.e., $[A^{(m)}] = [A^{(m)}]^{-1}$, or equivalently, $[A^{(m)}]^2 = [I]$, the identity under substitution, $[K] = [A^{(-1)}]$ and $[A] = [A^{(1)}]$ are conjugate duals as are $[N] = [N^{(1)}]$ and $[N]^{-1}=[N^{(-1)}]$. Note $[N]^{-m} = [N^{(-m)}] \neq [I]$, except for $m=0$, whereas $[A^{(m)}]^n = [I]$. Nicely, $[A^{(0)}] = [R]$, the reciprocal polynomials for multiplicative inversion of o.g.f.s., reducing to the (shifted) Pascal triangle row polynomials. $\endgroup$
    – Tom Copeland
    Commented Mar 19, 2023 at 15:08
  • $\begingroup$ Perhaps related: "Noncommutative analogs of monomial symmetric functions, Cauchy identity, and Hall scalar product." by Lenny Tevlin $\endgroup$
    – Tom Copeland
    Commented Jun 9, 2023 at 3:47

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