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Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-polynomials and h-polynomials for families of convex polytopes and simplicial complexes.

Ancillary question: Can someone point me to a reference illustrating the simplicial complexes for the low order tropical Grassmannians $G(2,n)$ or $T_n$ and their duals?

REGULAR f-h IDENTITIES:

For simple convex polytopes such as the associahedra and permutahedra the h-polynomials $h(t)$ and f-polynomials, or face polynomials, $f(t)$, satisfy the simple f-h identity $f(t) = h(1+t)$. This is also true of the dual of the simplicial complexes $G(2,n)$ of the tropical Grassmannians.

Examples:

1) The 3-D associahedron

The face polynomial (A033282 / A086810 / A126216) is

$f_{A_3}(t) = 14 + 21 t + 9 t^2 + t^3$

corresponding to 14 vertices (0-dimensional faces), 21 edges (1-D faces), 9 polygons (3 squares and 6 pentagons, 2-D faces), and 1 associahedron (3-D face), and the associated h-polynomial is the Narayana polynomial (A001263)

$h_{A_3}(t) = 1 + 6t + 6t^2 + t^3$,

and indeed $h_{A_3}(1+t) = f_{A3}(t)$ (quick check: $1+6+6+1 =14$, a Catalan number, A000108).

2) The 3-D permutahedron (an Archimedean polytope)

The face polynomial (A019538) is

$f_{P_3}(t) = 24 + 36 t + 14 t^2 + t^3$,

corresponding to 24 vertices (0-dimensional faces), 36 edges (1-D faces), 14 polygons (6 squares and 8 hexagons, 2-D faces), and 1 permutahedron (3-D face), and the associated h-polynomial is the Eulerian polynomial (A008292).

$h_{P_3}(t) = 1 + 11t + 11t^2 + t^3.$

3) Dual to the tropical Grassmannian $G(2,6)$ or $T_6$ (cf. this MO-Q)

(I don't have a clear picture of what these complexes look like, but there is some info on that in the OEIS entry. Perhaps someone could point me to a good ref on visuals.) The face polynomial is the Ward polynomial (A134991)

$f_{G(2,6)}(t) = 105 + 105 t + 25 t^2 +t^3$,

and the associated h-vector, a second-order Eulerian polynomial of A008517, is

$h_{G(2,6)}(t) = 24 + 58 t + 22 t^2 +t^3.$

REFINED f-h IDENTITIES

The regular f-h identities above are imposed by more refined relationships--an identity between the refined Euler characteristic partition polynomials and associated refined h-polynomials, an identity which naturally reduces to the f-h identity.

Particular instances:

Compare the specific examples above of the regular f- and h-polynomials with their refined counterparts.

1) 3-D associahedron: The refined Euler characteristic polynomial / signed refined face partition polynomials of the 3-D associahedron (cf. normalized A133437) is

$$A_4(u_1,u_2,u_3,u_4) = 14 u_1^4 - 21 u_1^2 u_2 + 6 u_1 u_3 + 3 u_2^2 - u_4$$

with the reduction $(14,21,(6+3),1) = (14,21,9,1)$,

and the associated refined h-polynomial is the refined Narayana / noncrossing-partitions polynomial (cf. A134264) is

$$N_4 = u_4 + + 2 u_2^2 + 4 u_3 h_1 + 6 u_2 u_1^2 + u_1^4$$

with the reduction $(1,(2+4),6,1) = (1,6,6,1)$.

2) 3-D permutahedron: The refined Euler characteristic polynomial / signed refined face partition polynomials of the 3-D permutahedron (cf. A133314) is

$$P_4 = 24 a_1^4 - 36 a_2 a_1^2 + 8 a_3 a_1 + 6 a_2^2 - a_4$$

with the reduction $(24,36,(8+6),1) = (24,36,14,1)$,

and the associated refined h-polynomial is the refined Eulerian partition polynomial (cf. A145271)

$$E_4 = a_1^4 + 11 a_1^2 a_2 + 4 a_2^2 + 7 a_1 a_3 + a_4$$

with the reduction $(1,11,(4+7),1) = (1,11,11,10)$.

3) Refined dual (?) to $G(2,6)$, or $T_6$: The refined Euler characteristic polynomial / signed refined face partition polynomials of the refined $G(2,6)$, or T_6, (cf. A134685) is

$$L_4 = 105 u_1^4 - 105 u_1^2 u_2 + 15 u_1u_3 + 10 u_2^2 - u_4$$

with the reduction $(105,105,(15+10),1) = (105,105,25,1)$,

and the associated refined h-polynomial is the refined second-order Eulerian partition polynomial (A360947, under construction)

$$E^{(2)}_4 = 24 a_1^4 + 58 a_2 a_1^2 + 14 a_3 a_1 + 8 a_2^2 + a_4$$

with the reduction $(24,58,(14+8),1)= (24,58,22,1)$.

The general refined f-h identities:

For the associahedra: With the set of associahedra polynomials $[A]$ of A133437 (normalized and re-indexed, see examples below) for compositional inversion of e.g.f.s, the set of noncrossing partition polynomials/ refined Narayana polynomials $[N]$ / free cumulant polynomials of A134264, and the reciprocal polynomials $[R]$ / the refined Pascal polynomials of signed A263633 for multiplicative inversion of o.g.f.s, the refined f-h identity is, in succinct notation,

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

representing at the row polynomial level for an infinite set of arbitrary, commuting, independent indeterminates $u_n$

$$N_n(u_1,u_2,...,u_n)|_{u_k = R_k(u_1,...,u_k)} = N_n(R_1(u_1),R_2(u_1,u_2),...,R_n(u_1,...,u_n)) = A_n(u_1,u_2,...,u_n).$$

Conversely, since multiplicative and compositional inversions are involutions, i.e., $[R]^2 = [I] =[A]^2$, where $[I]$ is the identity for substitution, other identities follow, in particular,

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

reflecting $f(t) = h(1+t)$ is equivalent to $f(t-1) = h(t).$

The set of reciprocal polynomials $[R]$, characterizing the underlying combinatorics of multiplicative inversion and a refinement of the row polynomials $(1+t)^n$ of the Pascal matrix, serves as an intermediary between the refined f- and h-polynomials of the associahedra just as the row polynomials of the Pascal matrix do for the ordinary f- and h-polynomials.

For the permutahedra: With the Lagrange inversion polynomials $[L]$ of A134685 for compositional inversion of e.g.f.s, or Taylor series; the refined Eulerian polynomials $[E]$ of A145271; and the permutahedra polynomials $[P]$ of A133314 for multiplicative inversion of e.g.f.s, the refined f-h identity is

$$[L][P] =[E].$$

Since $[L]^2 = [I] = [P]^2$, also

$$[P] =[L][E].$$

$[L]$ is the intermediary between the refined h-polynomials of the permutahedra $[E]$ and the signed refined f-polynomials / refined Euler characteristic polynomials of the permutahedra $[P]$.

For the tropical Grassmannians or their dual simplicial complexes: With the Lagrange inversion polynomials $[L]$ of A134685 for compositional inversion of e.g.f.s, or Taylor series; the refined second-order Eulerian polynomials $[E]^2$ of A360947 (under construction); and the reciprocal tangent polynomials $[RT]$ of A356144 for multiplicative inversion of the derivative of e.g.f.s, the refined f-h identity is

$$[E]^2 = [L][P][E] = [L][RT].$$

Then $[RT]$ is the intermediary between the refined signed f-polynomials $[L]$ of the simplicial complexes associated with the tropical Grassmannians $G(2,n)$ and their associated refined h-polynomials $[E]^2$, the refined second-order Eulerian polynomials.

Two related papers by other researchers (neither fully encompasses the observations above although N & T present a formulation of a noncommutative case that is a potential further refinement of the refined f-h identity for associahedra):

"Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials" by Price and Sokal

"Noncommutative Symmetric Functions and Lagrange Inversion II: Noncrossing partitions and the Farahat-Higman algebra" by Novelli and Thibon

A compilation of the first several polynomials for $[P]$,$[L]$,$[RT]$, and $[E]$ are at my personal blog / mini-arXiv Shadows of Simplicity. Below are the first few for $[A],[N]$ and $[R]$.

$[A]$, the associahedra partition polynomials, normalized A133437, the coefficients of the formal compositional inverse

$O^{(-1)}(x) = x + A_1(u_1) x^2 + A_2(u_1,u_2) x^3 + \cdots$

of the ordinary generating function (o.g.f.) / power series

$O(x) = x + u_1 x^2 + u_2 x^3 + \cdots$.

The counterpart for exponential generating functions (e.g.f.s) / Taylor series is $[L]$.

$A_1(u_1) = -u_1$,

$A_2(u_1,u_2) = 2u_1^2-u_2$,

$A_3(u_1,u_2,u_3) = -5u_1^3+5u_1u_2 - u_3$,

$A_4(u_1,u_2,u_3,u_4) = 14 u_1^4 - 21 u_1^2 u_2 + 6 u_1 u_3 + 3 u_2^2 - u_4.$

$[R]$, the reciprocal partition polynomials, signed A263633, giving the shifted reciprocal of the o.g.f. $O(x)$ as

$\frac{x}{O(x)} = \frac{x}{x + u_1 x^2 + u_2 x^3 + \cdots} = 1 + R_1(u_1) x+ R_2(u_1,u_2)x^2 + \cdots$.

The counterpart for e.g.f.s is $[P]$ defined by $\frac{1}{1+u_1 x + u_2\frac{x^2}{2!}+\cdots}$ where, for use with $[E]$ when the compositional inverse (CI) of $f(x)$ is desired, the denominator is the derivative $f(x)$.

$R_1(u_1) = - u_1,$

$R_2(u_1,u_2)= u_1^2 - u_2, $

$R_3(u_1,u_2,u_3) = -u_1^3 + 2 u_1 u_2 - u_3,$

$R_4(u_1,...,u_4) = u_1^4 - 3 u_1^2 u_2 + 2 u_1 u_3 + u_2^2 - u_4.$

$[N]$, noncrossing partitions polynomials / refined Narayana polynomials / free cumulant polynomials of free probability theory giving the free moments, A134264, defined by

$O^{(-1)}(x) = x + N_1(R_1(u)) x^2 + N_2(R_1(u_1),R_2(u_1,u_2)) x^3 + \cdots.$

The counterpart for e.g.f.s for $N_n(R_1(u_1),...,R_n(u_1,...,u_n))$ is $E_n(P_1(\bar{u}_1),...,P_n(\bar{u}_1,...,\bar{u}_n))$.

$N_1 = u_1$,

$N_2 = u_2 + u_1^2$,

$N_3 = u_3 + 3 u_2 u_1 + u_1^3$,

$N_4 = u_4 + + 2 u_2^2 + 4 u_3 h_1 + 6 u_2 u_1^2 + u_1^4.$

Tom Copeland
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    Side question. If in $L_4$ we replace $u_i$ by the power-sum symmetric function $p_i$, then we obtain an $e$-positive symmetric function. Is there some reason for this? – Richard Stanley Mar 01 '23 at 02:31
  • @RichardStanley, first, definition / ref / example of a prototypical e-positive symmetric function? Second, is this true for $L_5$ and $L_6$ also? If so, very likely to be a sure bet in general. The paper by N&T deals with sym fcts. I think that you are probably quite familiar with the content and can tease out an answer from it far more readily than I could. (I associate the power sums with o.g.f.s, so that is a surprising cross-fertilization if true in general.) – Tom Copeland Mar 01 '23 at 03:30
  • The $T_4$ simplicial complex is illustrated in "Polyhedral Covers of Tree Space" by Devadoss, Huang, and Spadacene (https://arxiv.org/abs/1311.0766). The top dimensional simplices of $T_5$ are discussed in "Partition complexes, duality and integral tree representations" by Robinson (https://arxiv.org/pdf/math/0410555.pdf). $T_5$ is called $T_6$ and described on p. 8 in "The tropical Grassmannian" by Speyer and Sturmfels (https://arxiv.org/pdf/math/0304218.pdf). – Tom Copeland Mar 01 '23 at 08:06
  • @ Tom Copeland: for a context involving e-positivity, see for example the slides of Michelle Wachs at https://math.dartmouth.edu/~orellana/wachs. Also, where can I find $L_5$ and $L_6$? – Richard Stanley Mar 02 '23 at 15:21
  • @RichardStanley, Click on Shadows of Simplicity in the body of the question. For computational purposes, you can conveniently copy the second presentation of the polynomials in LaTex in the lower sections of the notes. – Tom Copeland Mar 02 '23 at 15:41
  • @RichardStanley, see my formulas dated Sep 18 2014 in A008292 relating the bivariate e.g.f. of the Eulerian polynomials to the complete homogeneous symmetric polynomials via compositional inversion. Maybe this hints at the more general conjectures. – Tom Copeland Mar 02 '23 at 20:11
  • I checked that $L_6$, $-L_7$, and $L_8$ are $e$-positive after substituting $p_i$ for $u_i$. This was after replacing a_1^2 a_1^3 with a_1^2 a_2^3 and a_2 a_4^2 with a_2 a_3^2. Was this correct? – Richard Stanley Mar 04 '23 at 18:30
  • @RichardStanley, let me check, I'm prone to typos, cut-paste errors, and transcription errors. Off-hand, my monomial a_2a_4^2 must be incorrect since it gives (12+24) = 10 rather than 8. Ditto for a_1^2a_1^3. (Thought I had done this quick sanity check!) – Tom Copeland Mar 04 '23 at 18:49
  • @ Tom Copeland: Defining $\deg a_i=i$, $L_n$ should be homogeneous of degree $n$. – Richard Stanley Mar 04 '23 at 19:13
  • Btw, your transformation is imposd by the power-sum plethysm operator, or Adams operator $\psi^m$, discussed in "Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions" by Novelli and Thibon., They present both coarser sets of mono-variable polynomials, the (1+q)-Narayana polynomials, corresponding to reductions of my $[A^{(m)}]$ defined in https://mathoverflow.net/questions/422539/infinite-dimensional-involutions-infinitely-large-sets-of-multivariate-polynomi , and more refined sets than mine with noncommutative indeterminates. Fig 10 is reduced $[A^{(2)}]$. – Tom Copeland Mar 04 '23 at 19:13
  • @RichardStanley, the monomials of $L_n$ with the indexing in my compilation correspond to partitions of $n$, so the sum of the products of the exponents and subscripts should give $n$, and, hence, I call them partition polynomials. For $L_8$, your correction a_2^1 a_3^2 gives, as it should, (21+32) = 8. This is the type of check I typically do, but somehow, through several revisions, I missed or re-introduced my errors. Thanks for alerting me. Will correct the compilation. (I usually sum the numerical coefficients as a check also.) – Tom Copeland Mar 04 '23 at 19:39
  • @Richard Stanley, The numerical coefficients of my $[L]$ do indeed sum to the big Schroeder numbers and agree with those compiled by Lang in https://oeis.org/A176740/a176740_1.pdf. Could you send a file, or photo, of your results to me at tccopeland at gmail? – Tom Copeland Mar 04 '23 at 20:58
  • "Enumeration of Stable Graphs" by Kane Dillon lists legs and edges of graphs enumerated by A134991 (https://etd.ohiolink.edu/apexprod/rws_etd/send_file/send?accession=bgsu1648498756736004&disposition=inline). – Tom Copeland Mar 05 '23 at 09:23
  • Erratum for my comment above: For checks, the sums of absolute value of coefficients of the polynomials of $[L]$, A134685, and the Ward polynomials, A134991, are given by the Schroeder numbers A000311 (different from the big and little Schroeder numbers). – Tom Copeland Mar 06 '23 at 04:50

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