One Matrix to Rule Them All

Posted on July 27, 2022 by Tom Copeland

Schur self-Konvolution expansion Koefficients; inversion of Laurent and power series; and associahedra, noncrossing, and reciprocal partition polynomials

Springboarding off the work of Lagrange, in the 1940s Schur introduced what I will refer to (mnemonically) as the Schur self-Konvolution expansion Koefficients defined by

(1 + c_1 \; x + c_2 \; x^2 +c_3 \; x^3 + \cdots)^p = 1 + \sum_{k \geq 1} K_n^p(c_1,c_2,\cdots,c_n)\; x^n.

Specializations of this matrix encompass some well-known sets of partition polynomials of much interest in algebra and analysis, combinatorics, topology and geometry, and physics.

Prescribe

h(x) = \frac{x}{f(x)} = \frac{1}{1+ \bar{c}_1 \;x + \bar{c}_2\; x^2+\cdots} = 1+c_1\; x + c_2 \;x^2 +c_3\; x^3+\cdots.

Then

\frac{K_{n-1}^n(c_1,\cdots,c_n)}{n} = \frac{D_{x=0}^{n-1}}{(n-1)!} \frac{(h(x))^n}{n}

= \frac{D_{x=0}^{n-1}}{(n-1)!} \frac{(\frac{x}{f(x)})^n }{n} = \frac{D_{x=0}^n}{n!}\;f^{(-1)}(x) = N_n(c_1,\cdots,c_n) = A(\bar{c}_1,\cdots,\bar{c}_n),

giving the coefficients of the power series / ordinary generating function (o.g.f.) that is the compositional inverse at the origin of the o.g.f. f(x), where the partition polynomials N_n(c_1,\cdots,c_n) are the refined Narayana polynomials of OEIS A134264, labeling and enumerating noncrossing partitions (among other constructs), and the (re-indexed) partition polynomials A_n(\bar{c}_1,\cdots,\bar{c}_n) are those of A133437, the refined Euler characteristic polynomials of the associahedra;

with

\frac{f(x)}{x} =\frac{1}{h(x)} = \frac{1}{1+c_1\; x + c_2 \;x^2 +c_3\; x^3+\cdots)} = \sum_{n \geq 0} R_n(c_1,\cdots,c_n) \; x^n = \sum_{n \geq 0}\bar{c}_n \; x^n ,

K_{n}^{-1}(c_1,\cdots,c_n) = \frac{D_{x=0}^n}{n!}\; \frac{1}{h(x)} =R_n(c_1,\cdots,c_n) = \bar{c}_n,

giving the reciprocal partition polynomials R_n(c_1,\cdots,c_n), the coefficients of the multiplicative inverse of the o.g.f. h(x), which are OEIS A263633 (mod signs); and

\frac{K_{n}^{n-1}(c_1,\cdots,c_n)}{n-1} =\frac{D_{x=0}^n}{n!} \frac{(h(x))^{n-1}}{n-1} =\frac{D_{x=0}^n}{n!} \frac{(\frac{x}{f(x))})^{n-1}}{n-1}= \frac{D_{x=0}^n}{n!}\;\frac{x}{f^{(-1)}(x) }= -b_n(c_1,\cdots,c_n),

giving the partition polynomials b_n of A355201, which provide both the coefficients of the o.g.f. that is the shifted reciprocal of the compositional inverse at the origin of f(x) and the coefficients of the Laurent series

LS^{(-1)}(z) = z + b_1 + \frac{b_2}{z} + \frac{b_3}{z^2} + \cdots,

which is the compositional inverse of the Laurent series

LS(z) = z + c_1 + \frac{c_2}{z} + \frac{c_3}{z^2} + \cdots.

I refer to these partition polynomials as the special involutive Schur self-convolution expansion coefficients, or simply the special Schur expansion coefficients.

Representing the substitution of one set of partition polynomials into another set as that set’s indeterminates using a matrix-type notation with [I] the identity transformation gives a concise rep of some of these relationships:

[R]^2 = [A]^2 = [b]^2 = [I],

[N] = [A][R],

and

[b] = [R][N] = [R][A][R],

a conjugation isomorphism.

Then also the inverse to the refined Narayana, or noncrossing partition, polynomials is the set

[N]^{-1} = [R][A] =[R][N][R] =[R]^{-1}[N][R] =[R][N][R]^{-1} = [b][R]

of A350499, and we have a conjugation isomorphism of a set of partition polynomials with its inverse set.

The set of notes (a pdf)

“One Matrix to Rule Them All: Schur self-Konvolution expansion Koefficients; inversion of Laurent and power series; and associahedra, noncrossing, and reciprocal partition polynomials“

goes into more detail and expands on these associations.

__________

Errata for the pdf:

Ignore the equation in the pdf “One Matrix to Rule Them All” just above the statement “Equation 5.53 on p. 38 of Stanley is”. It is not correct (probably an overlooked cut-and-paste-error) and isn’t used in any computations or derivations. Shortly afterwards the correct equation, the classic Lagrange inversion formula, is presented.

The first line in the section titled Reduction to the Narayana Polynomials, which reads “The associahedra partition polynomials ASP . . . ,” should read “The noncrossing partitions polynomials NCP . . . .”

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The equivalent formalism couched in terms of exponential generating functions (e.g.f.s) / formal Taylor series is intimately intertwined with the formalism of Appell Sheffer sequences (and Sheffer sequences in general) with the combinatorics of the permutahedra and phylogenetic trees playing critical roles (see the previous post). The refined Euler characteristic polynomials of the associahedra [A] are replaced by the classic Lagrange partition polynomials [L] of A134685, interpretable as weighted phylogenetic trees and other combinatorial constructs (perhaps as the refined Euler characteristic polynomials of the Whitehouse simplicial complex); the o.g.f. reciprocal polynomials [R], by the e.g.f. reciprocal polynomials [P] of A133314, a.k.a. the refined Euler characteristic polynomials of the permutahedra; and the refined Narayana polynomials [N], by the refined Eulerian polynomials [E] of A145271. They obey the identities


[P]^2 = [L]^2 = [I],


[E] = [L][P]


and a new set of partition polynomials, the counterpart of [b], can be defined as


[B] = [P][E] =[P][L][P],


of which the first few are


B_0 = 1,


B_1 = -a_1,


B_2 = a_1^2 - a_2, coefficients sum = to 2 (absolute values) or 0,


B_3 = - a_1^3 + 2 a_1a_2 - a_3, sums to 4 or 0,


B_4 = a_1^4 - 3 a_2 a_1^2 + a_3 a_1 + 2 a_2^2 - a_4,


reduces to (1,3,3,1) with the sum 8 or (1,-3,3,-1) with the sum 0,


B_5 =-a_1^5 + 4 a_2 a_1^3 - 2 a_3 a_1^2 - 4 a_2^2 a_1 - a_4 a_1 + 5 a_2 a_3 - a_5,


reduces to (1,4,6,6,1) with the sum 18 or (-1,4,-6,4,-1) with the sum 0.


[B] is involutive since [B]^2 = [P][L][P][P][L][P] = [I] just as [b]^2 =[I], and [B] gives the Taylor series coefficients of the reciprocal of the derivative of f^{(-1)}(x), i.e., 1 \; / \; \partial_x f^{(-1)}(x) in terms of those of 1/ \partial_x f(x) = 1 + \sum_{n \geq 0} \; a_n \frac{x^n}{n!}, in parallel with [b] giving the power series coefficients of the shifted reciprocal of the compositional inverse.

(Spot-check this last relation with the example B_n polynomials for f(x) = e^x-1 and \ln(1+x), and for f(x) = x/(1+x) and f^{(-1)}(x) = x/(1-x) for which 1/\partial_xf(x) = 1 + 2x +2 \frac{x^2}{2} and 1/\partial_xf^{(-1)}(x) = 1 - 2x +2 \frac{x^2}{2}. Another slightly more involved check is with the Catalan generating function \frac{1-\sqrt{1-4x}}{2} and its inverse f(x) = x-x^2.)

The inverse set of partition polynomials for the refined Eulerian polynomials is

[E]^{-1} = [P][L] = [P][E][P] = [P]^{-1}[E][P] = [P][E][P]^{-1} = [B][P] .

with examples given in the previous post. (Last equality corrected Aug, 8, 2022 to [P] rather than [E].)

Added Aug. 8, 2022: [B] = [RT] in A356144 (in draft). [E]^{-1} is A356145 (in draft).

Added Sep. 17, 2022: An MO-Q related to absent monomials in [B], or [RT], is “Outlier absences of monomials in a group of inversion partition polynomials“.

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  1. Pingback: As Above, So Below: (m)-associahedra and (m)-noncrossing partitions polynomials | Shadows of Simplicity

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