An Explicit Counterexample to Stanley's Rankwise Lower-Bound Conjecture for Differential Posets

2026-07-24

AI Summary

This work refutes Stanley’s conjecture that the lower bound on rank sizes in r-differential posets is achieved by the r-fold Cartesian power of Young’s lattice, denoted \( Y^r \). By means of explicit combinatorial constructions, we exhibit an infinite family of r-differential posets for every \( r \geq 3 \) whose fourth rank has cardinality strictly smaller than that of \( Y^r \), specifically satisfying \( |P_4| = |(Y^r)_4| - \lfloor r/3 \rfloor \). The key innovation lies in replacing certain lower cover sets at rank 4 in \( Y^3 \) and applying a reflection-extension technique to generate new differential poset structures. Notably, for \( r = 3 \), the first five rank sizes are \( 1, 3, 9, 22, 50 \), contrasting with \( 1, 3, 9, 22, 51 \) for \( Y^3 \), thereby disproving this long-standing lower-bound conjecture.

Abstract

In Problem~6 of his 1988 paper on differential posets, Stanley asked for the least possible cardinality of a fixed rank of an $r$-differential poset and suggested that the minimum should be attained by $Y^r$, the $r$-fold Cartesian power of Young's lattice. We disprove the resulting universal coefficientwise lower bound. For every $r\geq 3$, we construct an infinite $r$-differential poset $P^{(r)}$ satisfying \[ \card{P^{(r)}_4} =\card{(Y^r)_4}-\left\lfloor\frac r3\right\rfloor. \] For $r=3$, the construction replaces thirteen rank-four lower-cover blocks of $Y^3$ by twelve blocks with the same point and pair incidence multiplicities, producing the initial rank sequence $1,3,9,22,50$ instead of $1,3,9,22,51$. A reflection extension then yields an infinite differential poset. The construction does not address the cases $r=1$ and $r=2$.

Problem

• differential posets
• Stanley's conjecture
• rank cardinality
• Young's lattice
• lower-bound

Innovation

• differential posets
• Stanley's conjecture
• counterexample
• rank enumeration
• Young's lattice