Dear Doron and Kyle,
I thought of another approach to the gamma vector problem of Kyle's. It might be even more "book" than Doron's proof.
If f is a symmetric polynomial, let (\gamma_0,...,\gamma_m) be its gamma vector, and define
G_f = \sum \gamma_i x^i.
It is routine to check that G_{fg} = G_f G_g. It is also routine to check that the gamma vector of 1+x+...+x^k is given by
\gamma_i = (-1)^i \binom{k-i}{i}, 0 \leq i \leq [k/2].
Since \sum_i \binom{k-i}{i} = F_{k+1}, the proof follows.
The same reasoning can be generalized to the q-multinomial coefficients
\qbinom{n}{a_1,...,a_j} = F(q).
Write this as
(n)! = (a_1)! ... (a_j)! F(q),
where (i)! is the q-analogue of i!. We get that the sum of the absolute values of the entries of the gamma vector of F(q) is the multi-Fibonomial coefficient! Maybe this can be somehow used to construct a Fibonacci binomial poset as a kind of quotient or subposet of the lattice of subspaces of a vector space over F_q. See Exercise 3.196(b) of EC1.
Since \binom{k-i}{i} is the number of i-element subsets of {1,...,k-1} containing no two consecutive integers, we also see that if (\gamma_0,...) is the gamma vector of (n)!, then (-1)^i \gamma_i is the number of sequences (S_1,S_2,...,S_{n-2}), where S_j is a subset of {1,...,j} containing no two consecutive integers and
i = |S_1| + |S_2| + ... + |S_{n-2}|.
This answers the question of my previous email.
Best regards,
Richard