# When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?

Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta$. Recall that the $f$-vector of $\Delta$ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where $f_{-1}=1$ (counting the empty set as the unique $-1$-dimensional face). There are lots of known constraints on which vectors can arise as $f$-vectors for various classes of simplicial complexes.

Question 1: are there interesting classes $C$ of simplicial complexes where these known constraints on $f$-vectors in $C$ are sufficient to determine the $f$-vector $f(\Delta )$ of any complex $\Delta$ known to be an element of $C$ if what you are given is the unordered list of coordinates in $f(\Delta )$?

Now an analogous question for graded posets. Given a finite, graded poset $P$, recall that its rank-generating function is the polynomial $\sum_{i=0}^n a_i x^i$ where $a_i$ is the number of elements of rank $i$ in $P$ and $n$ is the rank of $P$.

Question 2: are there interesting classes of posets $P$ for which we can recover the rank-generating function from its unordered list of coefficients?

What motivated my two questions was thinking just a little about the following MathOverflow question:

Constructing a field from a spherical building

Specifically, a $d$-dimensional Tits building has the property that its vertices may be colored with $d+1$ colors so that no two vertices in a face have the same color. In the case of finite buildings, I wondered if counting how many vertices there are in each color class would provide enough information to recognize which color class is which. For instance, when the vertices are indexed by the subspaces of a finite vector space, then the colors indicate the dimensions of these subspaces. In this case, the numbers of vertices of the various colors will be the coefficients in the rank-generating function of a geometric lattice, so a positive solution to question 2 for geometric lattices would suffice to do this.

I haven't actually spent time trying to solve these questions, but rather just decided to throw these questions out here, since they were based on another MathOverflow question. I'd be interested either in classes with a positive answer or in ones where it can be demonstrated that this is not possible. It seemed like knowing unimodality would be helpful, for instance, but not enough by itself; there's also the issue in question 1 of being given the $f$-vector coordinates rather than the (more nicely behaved) $h$-vector coordinates.

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What about $f$-vectors of simplicial polytopes? The $g$-theorem puts some pretty strong conditions on these vectors that might be enough for the uniqueness you want. – Richard Stanley Nov 16 '12 at 1:32
Richard, Thank you for your thoughts on this. I was also wondering about complexes with convex ear decompositions, but will need to spend some time with this. By the way, one reason I was quite interested in the question I linked to above is that I've been trying to figure out how to do some things in a way that'd be sensitive to field characteristic, and it seemed like an answer to that question might help. – Patricia Hersh Nov 16 '12 at 14:07