Restricted partitions and q-Pell numbers

In this paper, we provide new combinatorial interpretations for the Pell numbers p_n in terms of finite set partitions. In particular, we identify six classes of partitions of size n, each avoiding a set of three classical patterns of length four, all of which have cardinality given by p_n. By restricting the statistic recording the number of inversions to one of these classes, and taking it jointly with the statistic recording the number of blocks, we obtain a new polynomial generalization of p_n. Similar considerations using the comajor index statistic yields a further generalization of the q-Pell number studied by Santos and Sills.