Let \(\{a_i\}^n_{i=1}\) be a sequence of natural numbers, \(0\leq a_i\leq n\) for \(i=1,\dots,n\), and \(A_{nm}\) be an \(n\times m\) array associated with this sequence, whose entries \(\alpha_{ij}=0,1\) such that \(\sum_{j=1}^m \alpha_{ij}=a_i\), \(i=1,\dots,n\), \(j=1,\dots,m\). A path of order \(k\) along \(A_{nm}\) is said to be a sequence of entries \(\alpha_{ij_1},\dots,\alpha_{nj_n}\) for which \(\sum_{i=1}^n \alpha_{ij_i}=k\), \(k=0,1,\dots,n\), \(j_ i=1,\dots,m\), \(i=1,\dots,n\). The authors deduce that the number of such paths in the case \(m=n\), denoted by \(g^k(a_,\dots,a_n;n),\) is equal to \((-1)^k\sum_{m=k}^n \binom{m}{k}s_n(n,n-m)n^{n-m}\) where \(s_j(j,i)\) is the generalized Stirling number of the first kind associated with the numbers \(a_1,\dots,a_j\): \((x-a_1)\dots(x-a_j)=\sum_{\ell =0}^j s_j(j,\ell)x^{\ell}\), introduced by \textit{L. Comtet} [C. R. Acad. Sci., Paris, Sér. A 275, 747--750 (1972; Zbl 0246.05006)]. Furthermore, the generating function of the numbers \(g^k(1,\dots,n;n)\) is obtained.