Language of words whose run lengths are all distinct Assume $ \Sigma=\{0,1\}$, is $L$ a regular language? If it is not, how should we prove it with pumping lemma?


  $$L = \{1^{a_1} 0^{a_2}\ldots 01^{a_k} \mid k \in \mathbb N , a_i \geq 0 , \text{ the $a_i$ are all different}\}  $$
The definition of your language is not completely clear, but whatever it is, if you consider
$$ L \cap 1^* 0^* = \{ 1^a 0^b \mid a \neq b \} $$
then you immediately see that $L$ is not regular.

If you have to prove it using the pumping lemma, then simply take a proof that works for the above language; it will work for $L$ as well.