Stanley Rank-Size Bounds for Differential Posets

Definitions and mathematical statement

Use the conventional definition of an $r$-differential poset: $r$ is a positive integer, and $P=\bigsqcup_{n\geq 0}P_n$ is a graded poset with finite ranks and a unique minimum element $\hat 0\in P_0$. Write $x\lessdot y$ when $y$ covers $x$. Each element covering exactly $k$ elements is covered by exactly $k+r$ elements. For any two distinct elements of the same rank, the number of their common lower covers equals the number of their common upper covers. Define \[ p_n=|P_n|,\qquad p_0=1. \] The supplied record does not explicitly give this definition or the domain of $r$; this conventional interpretation requires confirmation against the original report.

The research task is to strengthen, uniformly over all such posets, the supplied upper bound attributed in the curation to Byrnes: \[ p_n\leq 1+r\sum_{k=0}^{n-2}p_k+(r-1)p_{n-1},\qquad n\geq 2, \] in the direction of the following conjectural target, attributed to Stanley: \[ \text{For every integer }r\geq 1,\text{ every }r\text{-differential poset }P, \text{ and every integer }n\geq 2,\qquad p_n\leq r p_{n-1}+p_{n-2}. \] The range $n\geq 2$ is the natural range in which all displayed rank indices are nonnegative; the supplied record does not explicitly state the index range. The request to improve the first bound does not specify a particular intermediate inequality, a required amount of improvement, or an asymptotic criterion. Thus it is an open-ended research task with an explicit conjectural target, rather than a uniquely specified intermediate proposition. Proving the target would fulfill the requested direction of improvement, but the source also permits partial progress.

Short English statement

For an $r$-differential poset, let $p_n$ count its elements of rank $n$. Improve the supplied universal upper bound on $p_n$, with Stanley's conjectured bound $p_n\leq r p_{n-1}+p_{n-2}$ as the target. The source leaves the required degree of improvement unspecified; the conventional poset definition and parameter ranges above need source confirmation.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30003819$, entitled Stanley Rank-Size Bounds for Differential Posets, in the Oberwolfach Reports Open Problems collection. Exact source problem code: catalogue entry. The record supplies proposed year 2018 and names no proposer.
• Supplied report citation: Enumerative Combinatorics (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/23, pp. 1381--1464. https://doi.org/10.4171/owr/2018/23. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied curation attributes the conjectural target to Stanley and calls the existing estimate Byrnes' inequality; these attributions have not been independently checked here.
• Supplied literature source, labelled "original OWR report": https://doi.org/10.4171/owr/2018/23.
• Supplied literature source, labelled "rank gaps in differential posets": https://arxiv.org/abs/1806.03509.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open, says that the report presents Stanley's upper bound as open, and says that later work proves strict rank growth without proving this upper-bound conjecture or the requested improvement to Byrnes' inequality. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified, with an explanation that notation, typography, and sentence structure were repaired while preserving mathematical content. This supplied editorial label does not establish mathematical correctness or independent source verification.