Combinatorial properties of poly-Bernoulli relatives

In this note we augment the poly-Bernoulli family with two new combinatorial objects. We derive formulas for the relatives of the poly-Bernoulli numbers using the appropriate variations of combinatorial interpretations. Our goal is to show connections between the different areas where poly-Bernoulli numbers and their relatives appear and give examples how the combinatorial methods can be used for deriving formulas between integer arrays.