• DOI:10.1007/s00026-022-00600-8
• Corpus ID: 245837886

Partial Rank Symmetry of Distributive Lattices for Fences

@article{Elizalde2022PartialRS,
  title={Partial Rank Symmetry of Distributive Lattices for Fences},
  author={Sergi Elizalde and Bruce E. Sagan},
  journal={Annals of Combinatorics},
  year={2022}
}

• S. Elizalde, Bruce E. Sagan
• Published 9 January 2022
• Mathematics
• Annals of Combinatorics

Associated with any composition β = (a, b, . . .) is a corresponding fence poset F (β) whose covering relations are x1 ✁ x2 ✁ . . .✁ xa+1 ✄ xa+2 ✄ . . .✄ xa+b+1 ✁ xa+b+2 ✁ . . . . The distributive lattice L(β) of all lower order ideals of F (β) is important in the theory of cluster algebras. In addition, its rank generating function r(q;β) is used to define q-analogues of rational numbers. Oğuz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth and Sagan, which in turn implies a previous conjecture of Morier-Genoud and Ovsienko that r(q;β) is unimodal. We show that, when β has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of F (β) of size k equals the number of filters of size k, when k is below a certain value. Our proof is completely bijective. Oğuz and Ravichandran also introduced a circular version of fences and proved, using algebraic techniques, that the distributive lattice for such a poset is rank symmetric. We give a bijective proof of this result as well. We end with some questions and conjectures raised by this work.