Counterexample to Stanley's Rankwise Lower-Bound Conjecture for Differential Posets
A paper on arXiv disproves Richard Stanley's 1988 rankwise lower-bound conjecture for r-differential posets. For r ≥ 3, the authors construct an infinite r-differential poset P^(r) with |P^(r)_4| = |(Y^r)_4| - ⌊r/3⌋, where Y^r is the r-fold Cartesian power of Young's lattice. For r=3, the construction replaces thirteen rank-four lower-cover blocks of Y^3 with twelve blocks having the same point and pair incidence multiplicities, yielding initial rank sequence 1,3,9,22,50 instead of 1,3,9,22,51. A reflection extension gives an infinite differential poset. The cases r=1 and r=2 are not addressed.
Key facts
- Stanley's 1988 paper Problem 6 asked for least cardinality of a fixed rank of an r-differential poset.
- Stanley suggested minimum attained by Y^r, the r-fold Cartesian power of Young's lattice.
- The paper disproves the universal coefficientwise lower bound.
- Construction works for every r ≥ 3.
- For r=3, initial rank sequence is 1,3,9,22,50 instead of 1,3,9,22,51.
- Thirteen rank-four lower-cover blocks of Y^3 replaced by twelve blocks.
- Reflection extension yields infinite differential poset.
- Cases r=1 and r=2 are not addressed.
Entities
Institutions
- arXiv