• 1 Department of Mathematics

For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the $\textit{order}$ of the first entries is independent of the $\textit{order}$ of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated.