Power compositions and semi-Pell compositions

Abstract

In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-\( m \)-Pell compositions. We find that these are in bijection with certain weakly unimodal \( m \)-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects. In the special case of \( m = 2 \), we have a new combinatorial interpretation of the semi-Pell sequence and connections to other objects.