This paper establishes a lower bound on the number of states necessary in the worst case to simulate an n-state two-way nondeterministic finite automaton (2NFA) by a one-way unambiguous finite automaton (UFA). It is proved that for every n, there is a language recognized by an n-state 2NFA that requires a UFA with at least ∑k=1n(k-1)!k!nkn+1k = Ω(n2n+2/e2n) states, where nk denotes Stirling’s numbers of the second kind. This result is proved by estimating the rank of a certain matrix, which describes every possible behaviour of n-state 2NFAs during their computation.