The run transform

We consider the transform from sequences to triangular arrays defined in terms of generating functions by f ( x ) → 1 − x 1 − x y f ( x ( 1 − x ) 1 − x y ) . We establish a criterion for the transform of a nonnegative sequence to be nonnegative, and we show the transform counts certain classes of lattice paths by number of the so-called pyramid ascents and certain classes of partitions into sets of lists (blocks) by number of blocks that consist of increasing consecutive integers.