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 $n\geq r\geq 1$ and being classically combinatorial enumerators with $J_{n}^{\left( 1\right) }=J_{n}$. From the calculation by induction of $J_{\lambda }$ for $n\leq 6$, 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 $J_{\left( 3,2,1\right) }$ shows the existence of an obstacle, if one wants to prove the conjectures by application of Huh's theorem cited above.