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Partial Rank Symmetry of Distributive Lattices for Fences

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Abstract

Associated with any composition \(\beta =(a,b,\ldots )\) is a corresponding fence poset \(F(\beta )\) whose covering relations are

$$\begin{aligned} x_1\lhd x_2 \lhd \ldots \lhd x_{a+1}\rhd x_{a+2}\rhd \ldots \rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots . \end{aligned}$$

The distributive lattice \(L(\beta )\) of all lower order ideals of \(F(\beta )\) is important in the theory of cluster algebras. In addition, its rank generating function \(r(q;\beta )\) is used to define q-analogues of rational numbers. Kantarcı 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 \(r(q;\beta )\) is unimodal. We show that, when \(\beta \) has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of \(F(\beta )\) of size k equals the number of filters of size k,  when k is below a certain value. Our proof is completely bijective. Kantarcı 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.

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Authors and Affiliations

  1. Department of Mathematics, Dartmouth College, Hanover, NH,  03755-3551, USA

    Sergi Elizalde

  2. Department of Mathematics, Michigan State University, East Lansing, MI,  48824-1027, USA

    Bruce E. Sagan

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  1. Sergi Elizalde
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  2. Bruce E. Sagan
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Correspondence to Bruce E. Sagan.

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Elizalde, S., Sagan, B.E. Partial Rank Symmetry of Distributive Lattices for Fences. Ann. Comb. 27, 433–454 (2023). https://doi.org/10.1007/s00026-022-00600-8

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