Total positivity from a kind of lattice paths

Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The main purpose of this paper is to study total positivity of a matrix $M=[M_{n,k}]_{n,k}$ generated by the weighted lattice paths in $\mathbb{N}^2$ from the origin $(0,0)$ to the point $(k,n)$ consisting of types of steps: $(0,1)$ and $(1,t+i)$ for $0\leq i\leq \ell$, where each step $(0,1)$ from height~$n-1$ gets the weight~$b_n(\textbf{y})$ and each step $(1,t+i)$ from height~$n-t-i$ gets the weight $a_n^{(i)}(\textbf{x})$. Using an algebraic method, we prove that the $\textbf{x}$-total positivity of the weight matrix $[a_i^{(i-j)}(\textbf{x})]_{i,j}$ implies that of $M$. Furthermore, using the Lindstr\"{o}m-Gessel-Viennot lemma, we obtain that both $M$ and the Toeplitz matrix of each row sequence of $M$ with $t\geq1$ are $\textbf{x}$-totally positive under the following three cases respectively: (1) $\ell=1$, (2) $\ell=2$ and restrictions for $a_n^{(i)}$, (3) general $\ell$ and both $a^{(i)}_n$ and $b_n$ are independent of $n$. In addition, for the case (3), we show that the matrix $M$ is a Riordan array, present its explicit formula and prove total positivity of the Toeplitz matrix of the each column of $M$. In particular, from the results for Toeplitz-total positivity, we also obtain the P\'olya frequency and log-concavity of the corresponding sequence. Finally, as applications, we in a unified manner establish total positivity and the Toeplitz-total positivity for many well-known combinatorial triangles, including the Pascal triangle, the Pascal square, the Delannoy triangle, the Delannoy square, the signless Stirling triangle of the first kind, the Legendre-Stirling triangle of the first kind, the Jacobi-Stirling triangle of the first kind, the Brenti's recursive matrix, and so on.