Combinatorial properties of the numbers of tableaux of bounded height

We introduce an infinite family of lower triangular matrices ${\mathrm{\Gamma }}^{(s)}$, where ${\gamma }_{n,i}^s$ counts the standard Young tableaux on $n$ cells and with at most $s$ columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number ${\tau }_s(n)$ of tableaux on $n$ cells and with at most $s$ columns.