$f_k(t_1,\dots,t_k)$ is counting the number of integer points in the
"Pitman-Stanley polytope" $\Pi_k(t_1,\dots,t_k)$ defined <a
href="https://math.mit.edu/~rstan/papers/parkpoly.pdf">here</a>. The
notation $N(\Pi_k(\mathbf{t}))$ is used in this paper, which has the
determinantal formula given by Hugh Denoncourt, as well as a
combinatorial formula. The combinatorial formula is equivalent to
  $$ f_k(t_1,\dots,t_k)=\sum_{\mathbf{h}\in K_k} \binom{t_1+h_1}{h_1}
     \prod_{i=2}^k \binom{t_i+h_i-1}{h_i}, $$
where
  $$ K_k := \{\mathbf{h}\in\mathbb{N}^k\colon \sum_{i=1}^j h_i\geq
  j\ \mathrm{for\ all}\ 1\leq j\leq k-1\ \mathrm{and}\ \sum_{i=1}^k
  h_i=k \}. $$
The set $K_k$ has a Catalan number $C_k$ of elements.