Let $G$ be the (infinite) graph with vertex set $\mathbb{Z}^2$, and the following edges. When $x+y < 0$, the vertex $(x,y)$ has outgoing edges to $(x+1,y)$ and to $(x,y+1)$. When $x+y \geq 0$, the vertex $(x,y)$ has outgoing edges to $(x+1-k,y+k)$ for all $k \geq 0$. That is, to $(x+1,y)$, $(x,y+1)$, $(x-1,y+2)$, and so on. These vertices have infinite outdegree (and when $x+y>0$, infinite indegree) but we will only use finite subgraphs, with finite degrees.

Now the number of directed paths in $G$ from $(-a,0)$ to $(b+1-k,k)$ is equal to $S(a,b)$.

Indeed, each edge in $G$ increases the sum of coordinates $x+y$ by $1$. So every path from $(-a,0)$ to $(b+1-k,k)$ has length $a+b+1$. For a given path, label each of the $k$ steps by their vertical travel ($0$ for a step east, $1$ for a step north, $2$ for a step in direction $(-1,2)$, etc.). The total of the labels is $k$. The first $a$ steps have labels $0$ or $1$. Subsequent steps have labels $\geq 0$.

The pair $(-a,0)$, $(b+1-k,k)$ may be translated by $(m,-m)$ for any $m$.

In your original question you did not say exactly what determinant you are trying to evaluate. But at least some determinants of values $S(a,b)$ can now be interpreted as counting disjoint path systems in the graph $G$. Well, I don't know how easy it will be to count those paths, but anyway I hope it helps.