Poisson approximations on the free Wigner chaos

We prove that an adequately rescaled sequence {Fn} of self-adjoint operators, living inside a fixed free Wigner chaos of even order, converges in distribution to a centered free Poisson random variable with rate .>0 if and only if .(F4n).2.(F3n).2.2.. (where . is the relevant tracial state). This extends to a free setting some recent limit theorems by Nourdin and Peccati [Ann. Probab. 37 (2009) 1412.1426] and provides a noncentral counterpart to a result by Kemp et al. [Ann. Probab. 40 (2012) 1577.1635]. As a by-product of our findings, we show that Wigner chaoses of order strictly greater than 2 do not contain nonzero free Poisson random variables. Our techniques involve the so-called .Riordan numbers,. counting noncrossing partitions without singletons.