Examples of infinite dimensional involutions

Edit 2/25/23, as suggested by YCOR below: (Start)

The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford language definition of involution as

  1. MATHEMATICS: a function, transformation, or operator that is equal to its inverse, i.e., which gives the identity when applied to itself.

In this case the basic involution involves substitution into themselves of infinite sets, say $[P]=(P_1(u_1),P_2(u_1,u_2),P_3(u_1,u_2,u_3),...)$ or $[P]=(1,P_1(u_1),P_2(u_1,u_2),P_3(u_1,u_2,u_3),...)$, of homogeneous polynomials $P_n(u_1,u_2,...u_n)$ multivariate in the arbitrary, independent, commutative indeterminates $u_n$ and of degree no greater than $n$ to obtain the identity set under substitution $[I]=(u_1,u_2,u_3,...)$ or $[I]=(1,u_1,u_2,u_3,...)$, i.e., symbolically,

$[P] = [P]^{-1},$

or, equivalently,

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

representing the graded substitution operation for $n = 1,2,3,...$

$P_n(P_1(u_1),P_2(u_1,u_2),...,P_n(u_1,...,u_n)) = u_n$.

Below, in the question, I attach additional initial polynomials such as $MI_0(a_0)$ containing only $a_0,b_0,c_0,$ or $d_0$, but these indeterminates can be assigned a fixed value of $1$ and the initial polynomials ignored. In my answer, with a simpler construction, I retain the initial $1$ as a reminder that the initial constants or coefficients of the linear term in the associated o.g.f.s are $1$. (End)

I'm looking for more examples of involutions of the type portrayed below, in which two sets of indeterminates (real or complex) each can be transformed into each other using the same set of rational functions. What properties should I expect such involutions to share? (Except for the designated constraint on one indeterminate of one set, the other indeterminates in that set may be assigned arbitrary values if ultimate convergence of the associated power series is ignored.)

I) Involution related to multiplicative inversion of exponential generating functions (e.g.f.s):

Given a convergent Taylor series/e.g.f.

$f(z) = \sum_{n \geq 0} a_n \frac{z^n}{n!} = a_0 \sum_{n \geq 0} \frac{a_n}{a_0} \frac{z^n}{n!}$

with $a_0 \neq 1$, the reciprocal (multiplicative inverse) e.g.f

$f^{-1}(z) = \sum_{n \geq 0} b_n \frac{z^n}{n!}$

is given by an involution represented by the homogeneous rational functions described in OEIS A133314. The first few are

$b_0 = \frac{1}{a_0}[1] = MI_0(a_0)$

$b_1 = \frac{1}{a_0^2}[-a_1] = MI_1(a_0,a_1)$

$b_2 = \frac{1}{a_0^3}[-a_2a_0+2a_1^2 ] = MI_2(a_0,a_1,a_2)$

$b_3 =\frac{1}{a_0^4}[-a_3a_0^2+6a_2a_1a_0-6 a_1^3] = MI_3(a_0,a_1,a_2,a_3)$

$b_4 =\frac{1}{a_0^5}[-a_4a_0^3+8a_3a_1a_0^2+6a_2^2a_0^2-36a_2a_1^2a_0+24a_1^4 ] = MI_4(a_0,a_1,a_2,a_3,a_4) .$

The polynomial in the numerator of $MI_n(a_0,...,a_n)$ is homogeneous of order $n$ while the rational function $MI_n(a_0,...,a_n)$ is homogeneous of order $-1$, i.e., $MI_n(t\cdot a_0,...,t\cdot a_n) = t^{-1} \cdot MI_n(a_0,...,a_n).$ The set of rational functions is an involution in that $b_n = MI_n(a_0,...,a_n)$ and $a_n = MI_n(b_0,...,b_n).$ (With b_0 =a_0=1, the resulting polynomials are the refined Euler characteristics (or signed, refined face polynomials) of the permutahedra. The o.g.f. version is A263633, a refined version of the Pascal triangle.)

II) Involution related to compositional inversion of e.g.f.s:

Given the function

$h(z) = \sum_{n \geq 1} c_n \frac{z^n}{n!}$

with $c_1 \neq 0$, the compositional inverse is

$h^{(-1)}(z) = \sum_{n \geq 1} d_n \frac{z^n}{n!}$

with the first few coefficients given in A134685 as

$d_1 = \frac{1}{c_1} [ 1 ] = CI_1(c_1)$

$d_2 = \frac{1}{c_1^3} [ -c_2 ] =CI_2(c_1,c_2) $

$d_3 = \frac{1}{c_1^5} [ 3 c_2^2 - c_1c_3 ]=CI_3(c_1,c_2,c_3) $

$d_4 = \frac{1}{c_1^7} [ -15 c_2^3 + 10 c_1c_2c_3 - c_1^2 c_4 ]=CI_4(c_1,c_2,c_3,c_4) $

$d_5 = \frac{1}{c_1^9} [ 105 c_2^4 - 105 c_1c_2^2 c_3 + 15 c_1^2 c_2c_4 + 10 c_1^2 c_3^2 -c_1^3 c_5 ]=CI_5(c_1,c_2,c_3,c_4,c_5). $

Reducing the indices of the indeterminates by one gives the same partitions as in the first example in the numerators. The polynomial in the numerator of $CI_n(c_1,...,c_n)$ is homogeneous of order $n-1$ while the rational function $CI_n(c_1,...,c_n)$ is homogeneous of order $-n$, i.e., $CI_n(t\cdot c_1,...,t\cdot c_n) = t^{-n} \cdot CI_n(c_1,...,a_n).$ The set of rational functions is an involution in that $d_n = CI_n(c_1,...,c_n)$ and $c_n = CI_n(d_1,...,b_n).$ (The o.g.f version is A133437, the refined Euler characteristics (or signed, refined face polynomials) of the associahedra.)

(Proposition 2.16 on page 15 of "Hopf monoids and generalized permutahedra" by Marcelo Aguiar and Federico Ardila appears relevant.)

III) Involution related to compositional inversion of Laurent series

(Added June 10, 2022, from “Differential calculus on the Faber polynomials” by Helene Airault and Abdlilah Bouali based on a result by Schur, with $a_0,b_0$ included. Edited june 25, 2022: Dependency on $a_0$ corrected.)

Given the inverse pair of Laurent series

$f(z) = a_0 \; z + a_1 + \frac{a_2}{z} + \frac{a_3}{z^2} + \cdots$


$f^{(-1)}(z) = b_0 \; z + b_1 + \frac{b_2}{z} + \frac{b_3}{z^2} + \cdots,$

the first few equalities for the involution are

$b_0 = \frac{1}{a_0}$

$b_1 = -\frac{a_1}{a_0}$

$b_2 = -a_2$

$b_3 = -(a_1a_2+a_0a_3)$

$b_4 = - (a_1^2a_2 +a_0a_2^2 + + 2 a_0a_1 a_3+a_0^2 a_4)$

$b_5 = -( a_1^3 a_2+ 3 a_0a_1 a_2^2+ 3 a_0a_1^2 a_3+ 3a_0^2 a_2 a_3+3 a_0^2a_1 a_4 + a_0^3 a_5)$.

For example, for the first four grades of involution,

$b_0 = \frac{1}{a_0} = \frac{1}{\frac{1}{b_0}} =b_0$

$b_1 = -\frac{a_1}{a_0} = -\frac{-(b_1/b_0)}{1/b_0} = b_1 $

$b_2 = -a_2 = -(-b_2)= b_2$

$b_3 = -(a_1a_2+a_0a_3) = -[ -(\frac{b_1}{b_0})(-b_2) + \frac{1}{b_0}((-(b_1b_2+b_0b_3))] = b_3 .$

  • 1
    $\begingroup$ Maybe the involution $\omega$ on the ring $\Lambda$ of symmetric functions which swaps the elementary $e_n$ and complete homogeneous $h_n$ symmetric functions would give another natural example. $\endgroup$ May 14, 2022 at 19:54
  • $\begingroup$ Sorry, maybe that is exactly the same as your example #1. $\endgroup$ May 14, 2022 at 20:41
  • 1
    $\begingroup$ @SamHopkins, the o.g.f. version of example I, A263633 as noted in the example, covers the elementary and complete homogeneous functions (see my related comment in that OEIS entry.) $\endgroup$ May 14, 2022 at 20:43
  • $\begingroup$ @SamHopkins: Certainly related to the Bell polynomials and similar compositional partition polynomials, but the CPPs don't provide an involution since $f(g(x))$ is not typically the same as $g(f(x))$. $\endgroup$ May 14, 2022 at 20:50
  • $\begingroup$ @SamHopkins: The relations among the LIF, the LPT, and compositional polynomials such as the Bell partition polynomials, OEIS A036040, are given in my pdf "Lagrange a la Lah" (2011). The partition polynomials that are inverse to the Bell are noted in the OEIS entry. $\endgroup$ May 17, 2022 at 17:32

1 Answer 1


Exploring the algebra of multiplicative and compositional inversion (series inversion) for formal power series / ordinary generating functions (o.g.f.), I came across an infinite number of sets of involutive partition polynomials.

Edit, Mar 3, 2023, due to a hasty transcription of notes--(too many pots on the burners). Superscripts added and a few exponents corrected, conforming to item 11 in the MO-Q "Guises of the noncrossing partitions":


Explication of notation and implied identities:

For $m = 1,2,3,\cdots$, given the o.g.f.,

$O^{(m)}(x) = x + c_1x^{m+1} + c_2 x^{2m+1} + c_3 x^{3m+1} + \cdots$

define the infinite set of reciprocal partition polynomials

$[R] = (1,R_1(u_1),R_2(u_1,u_2),R_3(u_1,u_2,u_3),...)$

as the expansion coefficients of the shifted reciprocal

$h(x) = \frac{x}{O^{(1)}(x)} = \frac{1}{1+c_1x +c_2 x^2 +\cdots} = 1 + R_1(c_1) x + R_2(c_1,c_2) x^2 + \cdots.$

The $m$-associahedra polynomials

$[A^{(m)}] = [1,A^{(m)}_1(u_1),A^{(m)}_2(u_1,u_2),...]$

of the expansion of the compositional inverse $(O^{(m)}(x))^{(-1)}$ of $O^{(m)}(x)$ are defined by

$(O^{(m)}(x))^{(-1)} = x + A^{(m)}_1(c_1)x^{m+1} + A^{(m)}_2(c_1,c_2)x^{2m+1} + \cdots.$

The $m$-noncrossing-partitions polynomials (name justified below)

$[N^{(m)}] = [1,N^{(m)}_1(u_1),N^{(m)}_2(u_1,u_2),...]$

are defined as the coefficients of the compositional inverse $(O^{(m)}(x))^{(-1)}$ in terms of the shifted reciprocals

$(O^{(m)}(x))^{(-1)} = x + A^{(m)}_1(c_1)x^{m+1} + A^{(m)}_2(c_1,c_2)x^{2m+1} + \cdots$

$ = x + N^{(m)}_1(h_1)x^{m+1} + N^{(m)}_2(h_1,h_2)x^{2m+1} + N^{(m)}_3(h_1,h_2,h_3) x^{3m+1} + \cdots$

$ = x + N^{(m)}_1(R_1(c_1))x^{m+1} + N^{(m)}_2(R_1(c_1),R_2(c_1,c_2))x^{2m+1} + \cdots$;

that is, in the arbitrary, independent indeterminates $u_n$, $A^{(m)}_n(u_1,...,u_n)$ is given by the substitution of $R_k(u_1,...,u_k)$ for $u_k$ in $N^{(m)}_n(u_1,...,u_n)$, i.e.,

$A^{(m)}_n(u_1,u_2,...,u_n) = N^{(m)}_n(R_1(u_1),R_2(u_1,u_2),...,R_n(u_1,...,u_n)).$

Denote this substitution operation for the complete sets by

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

Then since $[A^{(m)}]^2 = [A^{(m)}][A^{(m)}]=[A^{(m)}][A^{(m)}]^{-1} = [A^{(m)}]^{0} = [I] = [R][R] = [R]^2$ with $[I]$ the substitution identity, also

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

with the inverse substitutions given by

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

In particular, with the notational definitions $[A^{(1)}]=[A]$, $[N^{(1)}]=[N]$, and $[N^{(-m)}]=[N^{(m)}]^{-1}$, the inverse noncrossing-partitions polynomials are given by

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

New from notes:


$$[N]^m =[N^{(m)}] = [A^{(m)}][R],$$


$$[N][N]^m = [N]^{m+1} = [N^{(m+1)}] = [A^{(m+1)}][R].$$

If $[N][N] = [N]^{2} = [N^{(2)}]$, then the general relations are established, and this is true according to a theorem stated by Peter Bala in the OEIS. (My notes that I'm converting to a pdf contain a proof of this using the Schur identity mentioned below.) Therefore, the general results hinge on


$ = N^{(2)}_n(u_1,...,u_n) = N_{2n}(0,u_1,0,u_2,...,0,u_n)$,

$A^{(2)}_n(u_1,...,u_n) = A_{2n}(0,u_1,0,u_2,...,0,u_n)$,

and the 'similarity' property

$R_n(u_1,u_2,...,u_n) = R_{2n}(0,u_1,0,u_2,...0,,u_n)$.

This equivalence in $[N]$ of self-substitution to 'aeration-deaeration' generalizes in a simple fashion as noted below.


Via an identity of Schur noted in the MO-Q "Guises of the noncrossing partitions", an extension of the Lagrange inversion formula, it can be shown that the raising operation

$$[A^{(m+1)}]= [N][A^{(m)}] = [N]^{m+1}[R]$$

and the lowering operation

$$[A^{(m-1)}] = [N^{(-1)}][A^{(m)}] = [N]^{-1}[A^{(m)}] $$

are valid for $m = 0,\pm 1, \pm 2$, i.e., any integer, and, consequently, the $[A^{(m)}]$ are all involutions.

It also follows that

$$[N] = [A^{(m+1)}][A^{(m)}]$$


$$[N^{(-1)}] = [N]^{-1} =[A^{(m)}][A^{(m+1)}].$$

Notes on special cases:

$[A^{(1)}]=[A]$ are the normalized re-indexed partition polynomials of A133437, the refined Euler characteristic polynomials of the associahedra, with linear coefficient unity, also related to dissections of convex polygons and various families of tree, among othe combinatorial constructs (cf. the MO-Q "Guises of the associahedra").

$[A^{(0)}] = [R]$ are the signed partition polynomials, refined Pascal polynomials, of signed OEIS A263633.

$[A^{(-1)}] = [R][N] = [K]$ are the special Schur self-convolution expansion coefficients / partition polynomials of A355201.

$ [A^{(-2)}] = [R][N]^2$ has a natural reduction to A286784, related to the Feynman diagrams of yet another quantum model for interacting fermions.

$[N^{(1)}]=[N]$ are the partition polynomials of A134264, enumerating the noncrossing partitions, a.k.a. the free cumulant partition polynomials generating the free moments from the free cumulants of free probability theory, related also to Dyck lattice paths, trees, ... (cf. the MO-Q "Guises of the noncrossing partitions").

$[N^{(-1)}]=[N]^{-1}$ are the inverse noncrossing partition polynomials of A350499, a.k.a. the free moment partition polynomials generating the free cumulants from the free moments of free probability theory (cf. the MO-Q "Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory").

For $m \geq 1$, $[A^{(m)}]$ can be determined by the aeration-deaeration process

$A^{(m)}_n(u_1,u_2,...,u_n) = A_{mn}(0,0,...,u_1,0,0,...,u_2,0,0,...,u_n)$

with periodically placed swathes of $m-1$ zeros,

and, more surprisingly, so can $[N^{(m)}] = [N]^m$, i.e.,

$N^{(m)}_n(u_1,u_2,...,u_n) = N_{mn}(0,0,...,u_1,0,0,...,u_2,0,0,...,u_n).$

$[A^{(2)}]$ and $[N^{(2)}]$ (A338135) appear in the 1976 paper "Planar diagrams" by E. Brezin, Itzykson, Parisi, and Zuber with the indeterminates being connected Green functions in a quantum field model and in the recent arXiv paper "Connecting Scalar Amplitudes using The Positive Tropical Grassmannian" by Cachazo and Umbert. Even more refined partitions with the indeterminates being noncommutative are presented in "Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra" by Novelli and Thibon, which reduce to these partition polynomials.

With $u_1 = 1$ or $t$ and $u_k =0$ otherwise, the $[A^{(m)}]$ for $m \geq 1$ reduce to the Euler-Fuss-Catalan sequences of numbers (see, e.g., A001764).

With the above analysis, it can be proven that $N^{(3)}$ (A354622) naturally reduces to A173020; $N^{(2)}$ (A338135), to A120986 or A108767; $[N]$ (A134264) to A001263; $[N^{(-1)}] = [N]^{-1}$ (A350499), to A060693 or A088617; $[A^{(3)}]$, to A243663; $[A^{(2)}]$ (A359534) to A243662 or A102537; $[A^{(0)}]$ (signed A263633), to A007318; $[A^{(1)}]$ (normalized A133437 / A111785 ), to A033282 or A126216; $[A^{(-1)}]$ (A355201), to A001263; and $[A^{(-2)}]$, to A286784. These reductions are associated to numerous combinatorial constructs.

The reduction of the partition polynomials essentially involves reducing $[R]$ to the polynomials $(1\pm x)^n$ of the Pascal triangle A007318, showing that the identities $[A^{(m)}]= [N^{m}][R]$ are generalizations of a variant of the Dehn-Sommerville relations relating face polynomials, or f-polynomials $f(t)$, to h-polynomials $h(t)$ for, e.g., convex polytopes by $f(t) = h(1+t)$. E.g., the reduction of unsigned $[A]$ gives the face polynomials of the associahedra with that for the 3-D associahedron being $f_3(t) = 14+ 21t+9t^2+ 1 t^3$ with 14 vertices, 21 edges, 9 convex polygons (3 squares + 6 pentagons), and 1 associahedron, and the associated h-polynomial is $h_3(t) = f_3(t-1) = 1 + 6 t + 6 t^2 + t^3$, a Narayana polynomial, imposed by $[N] = [A][R]$, or, equivalently, $h_3(1+t) = f_3(t)$, imposed by $[N][R] = [A]$.

Historical note: The first few polynomials of $[A^{(1)}] = [A]$ for the inverse of the generic o.g.f. $O^{(1)}(x)$ and those for $[A^{(2)}]$, for odd o.g.f. $O^{(2)}(x)$ appear in a letter written by Isaac Newton in 1676 to Henry Oldenburg.

  • $\begingroup$ In "Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions" by Novelli and Thibon , variants of reduced $[A^{(m)}]$ in Figs. 7, 10, and 11 are called modified or reversed x = (1+q) Narayana triangles and variants of the reduced $[N^{(m)}]$ in Figs. 4 , 5 and 6, the m-Narayana triangles. For the diagonals or first columns, depending on the presentation of reduced $[A^{(m)}]$ , see "A combinatorial interpretation for n-ary trees for negative n" (mathoverflow.net/questions/441724/…). $\endgroup$ Mar 5 at 3:50

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