On a particular specialization of monomial symmetric functions

Let $m_{\lambda }$ be the monomial symmetric functions, $\lambda$ being an integer partition of $n\in {\mathbb{N}}^{\ast }$. For the specialization corresponding to the $q$-deformation of the exponential, we prove that each $m_{\lambda }$ is associated with a polynomial $J_{\lambda }\left(q\right)$ whose coefficients belong to $\mathbb{Z}$. $J_{\lambda }$ is a generalization of the case $\lambda =\left(n\right)$ for which $J_{\left(n\right)}=J_n$ is the enumerator of tree inversions. Some relations between $J_{\lambda }$ and $J_n^{\left(r\right)}$ are obtained, these $J_n^{\left(r\right)}$ having been defined algebraically in a previous work of the author for $n\ge r\ge 1$ and being classically combinatorial enumerators with $J_n^{\left(1\right)}=J_n$. From the calculation by induction of $J_{\lambda }$ for $n\le 6$, we conjecture that the coefficients of each $J_{\lambda }$ are strictly positive and log-concave. As a consequence of Huh’s Theorem on the $h$-vector of matroid complex it is shown that the coefficients of $J_n^{\left(r\right)}$ are strictly positive and log-concave, which gives a second argument in favor of these conjectures. It is also proven that the last $n-1$ coefficients of $J_{\lambda }$ are proportional to the first coefficients of column $n-r-1$ of Pascal’s triangle, $r$ being the length of $\lambda$. This is a third argument to state the conjectures. The calculation of $J_{(3,2,1)}$ shows the existence of an obstacle, if one wants to prove the conjectures by application of Huh’s theorem cited above.