Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)

Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature?

There are two sets of partition polynomials, not in the OEIS, that serve as the coefficients of the differential components of the raising ops for the set $[A]$ of partition polynomials that enumerate the topologically distinct faces of the associahedra and the dual set $[N]$ that enumerate noncrossing partitions / parking functions, the refined h-polynomials of the associahedra. The raising ops with a simple sign change also generate the sets of special Schur self-Konvolution expansion coefficients $[K] = [A^{(-1)}]$ and the inverse noncrossing partitions $[N^{(-1)}] = [N]^{-1}$. These four sets are intimately related to compositional inversion of pairs of power and Laurent series; expansions of self-convolutions and inverse self-convolution of power series; and the algebra, combinatorics, geometry, and physics associated with free probability theory and the Weyl-Coxeter group $A_n$. Intertwined with these paths of exploration is symmetric function theory.

The first few log associahedra partition polynomials, or generalized Zernike polynomials $[Z]$ (not in the OEIS), are

$Z_0 =1$

$LA_1 =Z_1 = -u_1$

$LA_2 =Z_2 = 3 u_1^2 - 2u_2$

$LA_3 =Z_3 = -10 u_1^3 + 12 u_1 u_2 - 3 u_3$

$LA_4 = Z_4 = 35 u_1^4 - 60 u_1^2 u_2 + 10 u_2^2 + 20 u_1 u_3 - 4 u_4,$

generated via the series $A(t)$ of the set $[A]$ of associahedra polynomials of A133437 (normalized, re-indexed, with initial indeterminate assigned a value of $1$)

$A(t) = 1 + A_1 t + A_2 t^2 + \cdots$

$=1-u_1 t +(2 u_1^2-u_2) t^2 + ( - 5 u_1^3 + 5 u_1 u_2 - u_3) t^3 + (14 u_1^4 - 21 u_1^2 u_2 + 6 u_1 u_3 + 3 u_2^2 - u_4) t^4+\cdots$

by

$\ln(A(t)) = LA_1 t + LA_2 t^2/2 + LA_3 t^3/3 +\cdots,$

so they are intimately related to the Newton identities of symmetric function theory (see the Faber polynomials A263916) and the Sheffer-Appell calculus.

A natural reduction with $u_k = x$ is essentially OEIS A253283 for the unsigned coefficients $(-1)^kk\binom{n}{k}\binom{-n}{k}$ of the set of orthogonal Zernike polynomials of order 1 (see details 1 and details 2), intimately related to the Legendre, Chebyshev, Gegenbauer, and Jacobi polynomials (see the Sheng and Shen ref in the OEIS and A097610) and found in interesting physical contexts, including optics and non-abelian gauge theory. The reduction is characterized as the black diamond product of $x^n$ and $x^{n+1}$ on p. 8 of "Web matrices: structural properties and generating combinatorial identities" by Dukes and White.

Edit April 3, 2023: (Start)

The first few ParPs of $[ILA]$, the inverse set to $[Z]=[LA]$, are

$ILA_1 = -u_1$

$ILA_2 = (3u_1^2 - u_2)/2!$

$ILA_3 = (- 16 u_1^3 + 12 u_1 u_2 -2u_3 ) / 3!$

$ILA_4 = (125 u_1^4 - 150 u_2 u_1^2 + 40 u_3 u_1 + 15 u_2^2 - 6 u_4) / 4!,$

which, when reduced become the first few shifted, signed polynomials of A220883 with interesting properties noted by Peter Bala and with combinatorial interpretations on p. 25 of "Duplicial algebras and Lagrange inversion" by Novelli and Thibon. I do not have a multinomial coefficient for these ParPs, but since Stanley provides one for $[LA]$ (see Background below), a generalized Chu-Vandermonde formula should provide it.

(End)

A third set of closely related set of partition polynomials (not in the OEIS) are the 'log Narayana partition polynomials"

$LN_1 = u_1$,

$LN_2 = u_1^2 + 2u_2$,

$LN_3 = u_1^3 + 6 u_2 u_1 + 3 u_3$,

$LN_4 = u_1^4 + 12 u_2 u_1^2 + 12 u_3 u_1 + 6 u_2^2 + 4 u_4,$

generated via the series $n(t)$ of the set $[N]$ of parking functions / refined Narayana / noncrossing partitions polynomials of A134264

$N(t) = 1+N_1t + N_2 t + N_3 t^3 \cdots$

$= 1 + u_1 t + (u_2 + u_1^2) t^2 + (u_3 + 3 u_1 u_2 + u_1^3) t^3 + (u_4 + 4 u_1 u_3+ 2 u_2^2 + 6 u_1^2 u_2 + u_1^4) t^4 +\cdots$

by

$\ln(N(t)) = LN_1 t + LN_2 t^2/2 + LN_3 t^3/3 + \cdots.$

$[LN]$ reduces with $u_k = x$ to essentially A132813, h-vectors for a cluster complex associated to the root system $B_n$, with reverse OEIS A103371.

Background

Log associahedra:

Richard Stanley in "Supplementary Exercises for Chapter 7 (symmetric functions) of Enumerative Combinatorics, vol. 2" (version of 28 March 2023) introduces the coefficients for the set of partition polynomials I refer to as $[Z]$. Exercise 133 (d) on pg. 53 of the update presents three identities expressing three core sets of partition polynomials (ParPs) central to symmetric function theory (SFT) in terms of the set $[N]$ of ParPs enumerating parking functions and the noncrossing partitions, a.k.a. the refined Narayana numbers. The core sets are $[E]$, the components of the elementary symmetric series $E(t) = 1 + e_1t + e_2t^2 +\cdots$; $[H]$, those of the complete homogeneous symmetric series $H(t)$; and $[P]$ for the power sum symmetric series. The first two are related as reciprocals by $H(t) =1/E(t)$, so the equivalent ParP substitution reps are $[H] = [R][E]$ and $[R][H] = [E]$, where $[R]$ is the set of reciprocal ParPs for multiplicative inversion of power series usually encountered as the proxy $w$ involution in SFT. (Be aware of sign differences in different presentations.)

$[N]$ (OEIS A134264) and its inverse $[N]^{-1}$ (A350499) pervade the lit on algebraic and geometric combinatorics associated with the Weyl-Coxeter group $A_n$ and the algebra and physics associated with free probability theory (see MO-Q and MO-Q), as do the dual involutive set for compositional inversion, the associahedra ParPs $[A]$ (normalized, re-indexed, A133437, see MO-Q). These are related algebraically via

$$[A] = [N][R]$$

and, equivalently, since $[A]^2 =[R]^2 = [I]$, the identity mapping under substitution,

$$[N]^{-1} = [R][A].$$

This last is equivalent to

$$N^{(-1)}(t) = 1/A(t),$$

where

$N^{(-1)}(t) = 1 + N^{(-1)}_1t + N^{(-1)}_2 t^2 +\cdots$

and

$A(t) = 1 + A_1t + A_2 t^2 +\cdots,$

so the machinery of symmetric function theory and the Sheffer-Appel calculus can be applied to determining relations among the sets of partition polynomials and their 'logarithms'.

The three identities in the update re-expressed as ParP substitution identities (substitute right into left) are

I)

$$[E] = ([A])[N][H]$$

II)

$$[P] = (D\; \ln[A])[N][H] =([Z])[N][H]$$

III)

$$[H] = ([N]^{-1})[N][H].$$

Stanley gives the multinomial coefficients (mod signs perhaps) of the three sets in parentheses.

The first sub rep encodes

$$([A])[N][H] = [A][A][R][H] = [R][H] = [E],$$

the third is obviously true and involves the inverse noncrossing ParPs $[N^{(-1)}] = [N]^{-1}$, which give the free cumulants in terms of the free moments of free probability theory.

The second identity is more complicated and requires some unpacking but has very interesting ramifications. It expresses an identity for the power sum symmetric series that is usually encountered in terms of determinants in SFT or as a logarithmic derivative. The power sums series is related to the the elementary symmetric series (see the Newton identities but be aware of sign differences) by

$$P(t) = D_t \ln(E(t)) = \sum_{n \geq} F_n(e_1,e_2,...,e_n) t^n,$$

where $[F]$ is the set of Faber polynomials (OEIS A263916) (again mod signs), so define the generalized Zernike polynomials of order 1 by

$$[Z] = D \ln[A] = [F][A],$$

implying, since $[A]^2 = [I]$,

$$[Z][A] = [F].$$

That is, locally,

$$Z_n(u_1,u_2,...,u_n) = F_n(A_1(u_1),A_2(u_1,u_2),...,A_n(u_1,...,u_n))$$

defined, more conventionally, by

$$D_t \; \ln(1 + A_1(u_1) t + A_2(u_1,u_2) t^2 + \cdots) = \sum_{n \geq 0} Z_n(u_1,...,u_n) t^n = Z(t).$$

Since $[A]^2 = [I]$, the middle identity reduces to the usual prescription for the power sums in terms of the elementary symmetric functions

$$(D\; \ln[A])[N][H] =[Z][N][H] = [Z][A][R][H] = [F][E] = [P].$$

Since the logarithmic derivative determines the differential part of the raising op for Sheffer Appell polynomial sequences, $[Z]$ is the set for a raising op / generating set for the the ParPs of $[A]$ and $[N]^{-1} = [N^{(-1)}]$; more precisely,

$$R_\bar{A} = x + \sum_{n \geq 0} Z_n(u_1,u_2,...,u_n) \frac{D_x^n}{n!} = x + Z(D_x),$$

giving

$$R_\bar{A} \; \bar{A}_n(u_1,...,u_n) |_{x=0} = \bar{A}_{n+1}(u_1,...,u_{n+1}) = R^n\;1|_{x=0}$$

where $\bar{A}_n = n!\; A_n$, and

$$R_{\bar{N}^{-1)}} = x - Z(D_x),$$

giving

$$R_{\bar{N}^{(-1)}} \; \bar{N}^{(-1)}_n(u_1,...,u_n) |_{x=0} = \bar{N}^{(-1)}_{n+1}(u_1,...,u_{n+1}) = R^n \; 1|_{x=0}.$$

In general, Appell sequences and raising and lowering ops have associated core combinatorics, equivalent reps of simply modified Pascal matrices, moment integrals, convolution integrals, Hirzebruch genera, Graves-Pincherle-Lie-Weyl algebras, and generalized derivatives. The raising op can be morphed into a recursion relation and related to the production matrices of Riordan matrix theory. The formalism of umbral inverse pairs can be applied as well with the identity

$$\sum_{k =0}^n N_k^{(-1)}A_{n-k} = \delta_n$$

at its core in this case. In addition, $D_{u_1}N^{(-1)}_n = n \;A_{n-1}$.

(Note $[LA]=[Z] = -[LN^{(-1)}]$.)

Log Narayana:

In parallel with arguments above, for the set of noncrossing partition polynomials $[N]$ and the special Schur convolution expansion coefficients $[K] = [A^{(-1)}]$ of A355201 (see MO-Q)

$$[N] = [R][K]$$

implies

$$N(t)K(t) = 1$$

so the associated raising ops are

$$R_N = x + D_{t = D_x} \ln(N(t)) = x + LN(D_x)$$

and

$$R_K = x + D_{t = D_x} \ln(K(t) )= x - LN(D_x).$$

At the core of these relationships is

$$\sum_{k=0}^n N_{k}K_{n-k} = \delta_n.$$

In addition, $D_{u_1}N_n = n\; N_{n-1}$.

(Note $[LN] = -[LK]$.)