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## ag.algebraic geometry – Examples of infinite dimensional involutions

**Examples of infinite dimensional involutions**

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 (egfs):**

Given a convergent Taylor series/egf

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

with $a_0 \neq 1$the reciprocal (multiplicative inverse) egf

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

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 $-1$ie, $MI_n(t\cdot a_0,…,t\cdot a_n) = t^{-1} \cdot MI_n(a_0,…,a_n).$ The set of rational functions is an involution in that $b_n = MI_n(a_0,…,a_n)$ other $a_n = MI_n(b_0,…,b_n).$ (With b_0 =a_0=1, the resulting polynomials are the refined Euler characteristics (or signed, refined face polynomials) of the permutahedra. The ogf version is A263633a refined version of the Pascal triangle.)

**II) Involution related to compositional inversion:**

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 like

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

$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 $-n$ie, $CI_n(t\cdot c_1,…,t\cdot c_n) = t^{-n} \cdot CI_n(c_1,…,a_n).$ The set of rational functions is an involution in that $d_n = CI_n(c_1,…,c_n)$ other $c_n = CI_n(d_1,…,b_n).$ (The above version is A133437the 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.)

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