Path counting and rank gaps in differential posets

TL;DR

This paper investigates the size gaps between consecutive ranks in differential posets, introducing a path-based projection operator to establish new lower bounds and analyze structural subcomponents.

Contribution

It extends Miller's result by proving that rank gaps are at least twice the differential parameter, and develops a path counting method to analyze these gaps.

Findings

Proved that $ ext{Δ} p_n ext{ } ext{≥} ext{ } 2r$ for differential posets.

Introduced a projection operator based on path counting in the Hasse diagram.

Derived stronger bounds when the poset contains many threads.

Abstract

We study the gaps $\Delta p_n$ between consecutive rank sizes in $r$-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen Miller's result that $\Delta p_n \geq 1$, which resolved a longstanding conjecture of Stanley, by showing that $\Delta p_n \geq 2r$. We also obtain stronger bounds in the case that the poset has many substructures called threads.