Multiple consecutive runs of multi-state trials: distributions of $(k_1,k_2,\dots ,k_{\mathrm{\ell }})$ patterns

The pattern $(k_1,k_2,\dots ,k_{\mathrm{\ell }})$ is defined to have at least $k_1$ consecutive $1$’s followed by at least $k_2$ consecutive $2$’s, $\dots$, followed by at least $k_{\mathrm{\ell }}$ consecutive $\mathrm{\ell }$’s. By iteratively applying the method that was developed previously to decouple the combinatorial complexity involved in studying complicated patterns in random sequences, the distribution of pattern $(k_1,k_2,\dots ,k_{\mathrm{\ell }})$ is derived for arbitrary $\mathrm{\ell }$. Numerical examples are provided to illustrate the results.