(This is a bit too long for a comment, though not exhaustive at all.)

This number, especially if you make appropriate changes of your notation to replace $<$ by $\le$ (replace $a_i$ by $a_i-i$, and $x_i$ by $x_i-i$, that is), admits an interpretation in terms of Young lattice (inclusion partial order on Young diagrams), or, equivalently, in terms of lattice paths below the graph of $i\to a_i$).

In addition to the binomial coefficient example (in these updated terms it is the number of Young diagrams inside a rectangle, or lattice paths inside a rectangle, so manifestly a binomial coefficient), the answer which is very well known applies to the diagram $(n,n-1,\ldots,1)$, when it is the $n$th Catalan number, and more generally, for $(kn,k(n-1),\ldots,k)$, when it is the $n$th Fuss-Catalan number. This altogether suggests that there might be some hook-length-kind formula formula which I am missing, and maybe this incomplete answer will make someone who knows that formula to explain it...

(This is a bit too long for a comment, though not exhaustive at all.)

This number, especially if you make appropriate changes of your notation to replace $< $ by $\le$ (replace $a_i$ by $a_i-i$, and $x_i$ by $x_i-i$, that is), admits an interpretation in terms of Young lattice (inclusion partial order on Young diagrams), or, equivalently, in terms of lattice paths below the graph of $i\to a_i$).

In addition to the binomial coefficient example (in these updated terms it is the number of Young diagrams inside a rectangle, or lattice paths inside a rectangle, so manifestly a binomial coefficient), the answer which is very well known applies to the diagram $(n,n-1,\ldots,1)$, when it is the $n$th Catalan number, and more generally, for $(kn,k(n-1),\ldots,k)$, when it is the $n$th Fuss–Catalan number. This altogether suggests that there might be some hook-length-kind formula formula which I am missing, and maybe this incomplete answer will make someone who knows that formula to explain it...