The Run Transform

TL;DR

The paper introduces a new sequence-to-array transform based on generating functions, characterizes when it preserves nonnegativity, and links it to counting lattice paths and ordered partitions with specific properties.

Contribution

It establishes a criterion for nonnegativity preservation and connects the transform to combinatorial counting of lattice paths and ordered partitions.

Findings

Provides a criterion for nonnegative sequence transform

Counts lattice paths by pyramid ascents

Counts ordered partitions by block structure

Abstract

We consider the transform from sequences to triangular arrays defined in terms of generating functions by f(x) -> (1-x)/(1-xy) f(x(1-x)/(1-xy)). We establish a criterion for the transform of a nonnegative sequence to be nonnegative, and we show that the transform counts certain classes of lattice paths by number of "pyramid ascents", as well as certain classes of ordered partitions by number of blocks that consist of increasing consecutive integers.