On the rank function of a differential poset

TL;DR

This paper explores the properties of $r$-differential posets, establishing connections to hypergraphs, demonstrating non-polynomial growth of their rank functions, and revealing limitations of the Interval Property conjecture within this context.

Contribution

It introduces a bijection between differential posets and hypergraphs, proves the existence of nonisomorphic differential posets with identical rank functions for $r \, \geq 6$, and shows the rank function's nonpolynomial growth.

Findings

Established a bijection with hypergraphs including finite projective planes.

Proved existence of nonisomorphic differential posets with same rank function for $r \geq 6$.

Showed the rank function grows faster than any polynomial, approaching Hardy-Ramanujan asymptotics.

Abstract

We study $r$-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, including the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if $r\geq 6$, of $r$-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collaborators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function $p_n$ of any arbitrary $r$-differential poset has nonpolynomial growth; namely, $p_n\gg n^ae^{2\sqrt{rn}},$ a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions.