We introduce a two-parameter deformation of the classical Poisson distribution from the viewpoint of noncommutative probability theory, by defining a (q, t) -Poisson type operator (random variable) on the (q, t) -Fock space Bl12 (See also BY06, AY20). From the analogous viewpoint of the classical Poisson limit theorem in probability theory, we are naturally led to a family of orthogonal polynomials, which we call the (q, t) -Charlier polynomials. These generalize the q-Charlier polynomials of Saitoh-Yoshida SY00a, SY00b and reflect deeper combinatorial symmetries through the additional deformation parameter t. A central feature of this paper is the derivation of a combinatorial moment formula of the (q, t) -Poisson type operator and the (q, t) -Poisson distribution. This is accomplished by means of a card arrangement technique, which encodes set partitions together with crossing and nesting statistics. The resulting expression naturally exhibits a duality between these statistics, arising from a structure rooted in generalized Fibonacci numbers. Our approach provides a concrete framework where methods in combinatorics and theory of orthogonal polynomials are used to investigate the probabilistic properties arising from the (q, t) -deformation.
Asai et al. (2025) studied this question.