$f_k(t_1,\dots,t_k)$ is counting the number of integer points in the "Pitman-Stanley polytope" $\Pi_k(t_1,\dots,t_k)$ defined here. 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 $$ K_k := \{\mathbf{h}\in\mathbb{N}^n\colon \sum_{i=1}^j h_i\geq j\ \mathrm{for\ all}\ 1\leq j\leq k-1\ \mathrm{and}\ \sum_{i=1}^k h_i=k \}. $$ The set $K_k$ has a Catalan number $C_k$ of elements.

$f_k(t_1,\dots,t_k)$ is counting the number of integer points in the "Pitman-Stanley polytope" $\Pi_k(t_1,\dots,t_k)$ defined here. 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.