Poisson type operators on the Fock space of type B and in the Blitvi{\'c} model

In \cite{Bia97} Biane proposed a new statistic on set partitions which he called \emph{restricted crossings}. In a series of papers \cite{Ans01,Ans04,Ans04b,Ans05} Anshelevich showed that this statistic is an essential tool to investigate stochastic processes on $q$-Fock space. In particular, Anshelevich constructed operators whose moments count restricted crossings and used these operators to develop a beautiful theory of noncommutative $q$-L\'{e}vy processes. In the present paper following Anshelevich we define gauge operators on $(\alpha,q)$-Fock and cumulants which are governed by statistics on partitions of type B. In addition we investigate this construction in the context of a model of Blitvi{\'c} model \cite{B12}, where some related but different combinatorial structures appear, and we explain their relation with $t$-free probability.