On the transition reduction problem for finite automata

We are interested in the problem of transition reduction of nondeterministic automata. We present some results on the reduction of the automata recognizing the language $L(E_n)$ denoted by the regular expression $E_n=(1+\varepsilon)... (2+\varepsilon)... (3+\varepsilon)... (n+\varepsilon)$. These results can be used in the general case of the transition reduction problem.