## In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms

In the earlier post Compositional Inverse Operators and Sheffer Sequences, I constructed relations among a generic power series, call it $f(x)=x \cdot H(-x)$, or ordinary generating function (o.g.f.), its compositional inverse $f^{(-1)}(x)= [xH(-x)]^{(-1)}$ and four sets of Sheffer polynomial sequences–two Appell sequences $p_n(x)$ and $q_n(x)$ and two binomial Sheffer sequences $u_n(x)$ and $v_n(x)$ intimately related by

$e^{tp.(x)}= \frac{t}{tH(-t)} e^{xt}$,

$e^{tq.(x)}= \frac{[tH(-t)]^{(-1)}}{t} e^{xt}$,

$e^{tu.(x)}= e^{x \cdot tH(-t)}$,

$e^{tv.(x)}= e^{x \cdot [tH(-t)]^{(-1)}}$.

Tis method to my madness of writing these relations in a roundabout way by letting $f(t)=tH(-t)$. I want to find relations between the two families of symmetric polynomials— the elementary symmetric polynomials (ESPs), denoted by $e_n$, and the complete homogeneous symmetric polynomials (CSPs), denoted by $h_n$, whose o.g.f.s $H(t)$ and $E(t)$ are essentially reciprocal to each other:

$E(t) = \frac{t}{tH(-t)}=1+e_1t+e_2t^2+...=1/[1-h_1t+h_2t^2+...]$.

Nowhere will I use the explicit definitions of these symmetric polynomials so the relations uncovered will apply to basically any generic o.g.f. and the polynomial sequences formed from it and its shifted multiplicative and compostional inverses.

With these definitions the generating functions for the Appell sequence for $p_n(x)$ become

$e^{tp.(x)}= E(t) e^{xt} = exp[(\hat{e}. + x)t]$,

so $p_n(x) = (\hat{e}.+x)^n = \sum_{k=0}^n \binom{n}{k} \hat{e}_{n-k}x^k$,

where $\hat{e}_n = n! e_n$. The first few polynomials are

$p_0(x)=1$,

$p_1(x)= e_1 + x$

$p_2(x)=2e_2 + 2e_1x + x^2$,

$p_3(x)=6e_3 + 6e_2x + 3e_1x^2 + x^3$.

The numerical coefficients are given by OEIS A094587 and A008279.

The other sequence of Appel polynomials depend on the coefficients of the power series for the multiplicative inverse of $[tH(-t)]$ which is from A134264 which gives the multiplicative inverse in terms of the coefficients of the power series of the functions shifted reciprocal $t/[tH(-t)] = E(t) = 1+e_1t+e_2t^2+ \cdots$ as the signed refined partition polynomials for the noncrossing partitions $NC_n$

$[tH(-t)]^{-1}= NC_1t + NC_2t^2+ \cdots$

with the first few partition polynomials

$NC_1 = 1$,

$NC_2 = NC_2(1,e_1) = e_1$,

$NC_3 = NC_3(1,e_1,e_2) = e_1^2 + e_2$,

$NC_4 = NC_4(1,e_1 ,e_2,e_3) = e_1^3 + 3e_1e_2+e_3$.

An alternative is to use the inversion scheme of A133437 involving the signed refined face partition polynomials of the Stasheff polytopes, or associahedra. Then

$[tH(-t)]^{(-1)}= Assc_1t + Assc_2t^2 + \cdots$

with the first few polynomials

$Assc_1 = Assc_1(1)=1$,

$Assc_2 = Assc_2(1,-h_1) = h_1$

$Assc_3 = Assc_3(1,-h_1,h_2) = 2h_1^2-h_2$,

$Assc_4 =Assc_4(1,-h_1,h_2,-h_3) = 5h_1^3-5h_1h_2+h_3$.

OEIS A263633 can be used to convert between the ESPs and CSPs and so also between the $NC_n$ and $Assc_n$ .

The first few conversions are

$h_1=e_1$,

$h_2 = e_1^2 -e_2$,

$h_3 = e_1^3 - 2e_1e_2 + e_3$.

(The relation is an involution satisfied when $e_n$ and $h_n$ are interchanged.)

$\frac{[tH(-t)]^{(-1)}}{t}e^{xt}= e^{t\widehat{NC}.}e^{xt}=e^{(\widehat{NC}+x)t}$.

Then

$q_n(x)=(\widehat{NC}.+x)^n$

where $\widehat{NC}_n = n!NC_{n+1}$. The first few are

$q_0(x)=1$,

$q_1(x)= NC_2 + x= e_1+x$

$q_2(x)=2NC_3 + 2NC_2x + x^2=2(e_1^2+e_2)+e_1x+x^2$,

$q_3(x)=6NC_4 + 6NC_3x + 3NC_2x^2 + x^3$.

Now for the two binomial Sheffer sequences, we will use the refined Lah partition polynomials described in the Lagrange a la Lah pdf (Part I, cf. also A130561, these are the normalized, signed elementary Schur polynomials) with the e.g.f.

$\exp[ t \frac{c.x}{c.x-1}] = \exp[-t [c_1x+c_2x^2+...]]$.

The first few polynomials are

$Lah_0(t)=1$,

$Lah_1(c_1;t)=-c_1t$,

$Lah_2(c_1,c_2;t)= -2c_2 t+c_1^2t^2$,

$Lah_3(c_1,c_2,c_3;t)= -6c_3t+6c_2c_1t^2-c_1^3t^3$,

$Lah_4(t)= -24c_4t+(24c_1c_3+12c_2^2)t^2-12c_1^2c_2t^3+c_1^4t^4$.

Then

$u_n(t)=Lah_n(-1,h_1,-h_2,...;t)$

with the first few being

$u_0(t)=1$,

$u_1(t)= t$,

$u_2(t)=-2h_1t+t^2=-2e_1t+t^2$,

$u_3(t)=6h_2t-6h_1t^2+t^3=6(e_1^2-e^2)t-6e_1t^2+t^3$.

Similarly,

$v_n(t)= Lah_n(-NC_1,-NC_2,...;t)$

with the first few being

$v_0(t)=1$,

$v_1(t)=NC_1t= t$,

$v_2(t)=2NC_2t+NC_1^2t^2= 2e_1t+t^2$,

$v_3(t)=6NC_3t+6NC_2NC_1t^2+t^3=6(e_1^2+e_2)t+6e_1t^2+t^3$.

(Recall that converting the ESPs into CSPs gives products of $Assc_n$ also.)

The earlier post establishes the umbral similarity transformations (the row by row equivalent of a matrix similarity transformation)

$p_n(t) = u_n(q.(v.(t)))$

and

$q_n(t) = v_n(p.(u.(t)))$.

Let’s do some spot checks. First,

$p_1(t)=e_1+t$

and

$u_1(q.(v.(t))) = (q.(v.(t)))^1=q_1(v.(t))=e_1+v_1(t)=e_1+t$,

which is consistent.

Second check:

$p_2(t)=2e_2 + 2e_1t + t^2$,

and

$u_2(q.(v.(t))) =-2h_1q_1(v.(t))+q_2(v.(t))$

$=-2h_1[e_1+v_1(t)]+2(e_1^2+e_2)+2e_1v_1(t)+v_2(t)$

$=-2e_1[e_1+t]+2(e_1^2+e_2)+2e_1t+2e_1t+t^2=2e_2+2e_1t+t^2 = p_2(t)$.

With $t=0$, we obtain, for $n>0$,

$\hat{e}_n= n!e_n = p_n(0)=u_n(q.(0))=Lah_n(-1,h_1,-h_2, ...;a.)$

where $(a.)^n = a_n= n!NC_{n+1}=n!Assc_{n+1}$.

For example,

$u_3(q.(0))= 6h_2q_1(0)-6h_1q_2(0)+q_3(0)$

$= 6h_2NC_2-6h_1(2NC_3)+6NC_4$

$= 6(e_1^2-e_2)e_1-12e_1(e_1^2+e_2)+ 6(e_1^3+3e_1e_2+e_3)=6e_3=p_3(0)$,

and, substituting $Assc_n$ for $NC_n$,

$u_3(q.(0))=6h_2Assc_2-6h_1(2Assc_3)+6Assc_4$

$=6h_2(h_1)-12h_1(2h_1^2-h_2)+6(5h_1^3-5h_1h_2+h_3)$

$=(6+12-30)h_1h_2+(-24+30)h_1^3+6h_3=3!(h_1^3-3h_1h_2+h_3)= 3!e_3=p_3(0)$.

Reprising, the refined Lah polynomials (aka, the elementary Schur polynomials) can be used to generate the coefficients of the reciprocal, or multiplicatve inverse,

$E(t)=1+e_1t+e_2t^2+...$

of an o.g.f.

$H(-t)=1-h_1t+h_2t^2-...$

from the coefficients of the power series of the shifted o.g.f.’s compositional inverse

$(tH(-t))^{(-1)}=t\bar{H}(-t)=t(1-\bar{h}_1t+\bar{h}_2t^2-...)$

$= Assc_1(1)t+Assc_2(1,-h_1)t^2+Assc_3(1,-h_1,h_2)t^3+...$

$= NC_1(1)t+NC_2(1,e_1)t^2+NC_3(1,e_1,e_2)t^3+...$.

(Note that $h_n = Assc_n(1,-\bar{h}_1,\bar{h}_2,...)$ could be substituted into the above equations.)

This is a rather roundabout method to determine reciprocals when A263633 could be used more efficiently given the $h_n$, but it establishes transformations among partition polynomial sequences that encode information about important combinatorial constructs–permutations, associahedra, noncrossing partitions–and operations in analysis–multiplicative and compositional inversion.

The associations via the refined Lah polynomials are by virtue of the fact that the pair of Appell polynomials (which are not an inverse pair under umbral composition) are related to each other by a similarity transformation by an inverse pair of lower triangular matrices with the row polynomials $u_n(x)$ and $v_n(x)$, which are an inverse pair under umbral composition of the row polynomials, i.e., $u_n(v.(c.))= (c.)^n=c_n=v_n(u.(c.))$. Since these Appell polynomials can be generated by differential raising operators, these results can be recoded in terms of a pair of Appell raising operators or differential transform operators that transform each $x^n$ into an Appell polynomial.

Again, nowhere have explicit constructions of the ESPs or CSPs been required, so the formalism above applies to a generic o.g.f. $g(t)=1+g_1t+g_2t^2+...$ substituted for $E(t)$ or $H(t)$.

Finally, we find another of my favorite families of convex polytopes–the permutahedra–if we relate $\hat{e}_n$ to $\hat{h}_n$ via the signed, refined face partition polynomials (cf. A133314) of the permutahedra.

Edit (Oct. 1, 2019):

In “Topics in topology. Fall 2008: The signature theorem and some of its applications” by Liviu I. Nicolaescu, we find on page 85 two generating series related to L genuses and multiplicative sequences,$a^{\gamma}$ and $R$, to which the above formalism can easily be applied since they are related by

$a^{\gamma}(t^2) = \frac{t}{R^{(-1)}(t)}$ and $R(t) = \int_0^t r^\gamma(x)dx$

where $r^\gamma = 1 + \sum_{k > 0} \gamma(CP^{2n}) t^{2n}$.

OEIS A133932 could be used to compute $R^{(-1)}$ directly from the coefficients $\gamma(CP^{2n})$.