In a previous paper, we defined the notion of inset. In this paper, we first derive a generating function for the number of insets in terms of one of its parameters. Using this function, we connect insets with some important classes of integers.

We first prove that the numbers of integer partitions satisfy a system of homogeneous linear equations. Then we derive an explicit formula for the coefficients of the Euler product function in terms of the number of insets. As a consequence, we express the Euler pentagonal number theorem in terms of insets.

Finally, we derive an explicit formula for the entries of the Mahonian triangle.