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Associated with any composition beta=(a,b,...) is a corresponding fence posetF(beta) whose covering relations are  x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... Thedistributive lattice L(beta) of all lower order ideals of F(beta) is importantin the theory of cluster algebras. In addition, its rank generating functionr(q;beta) is used to define q-analogues of rational numbers. Oguz andRavichandran recently showed that its coefficients satisfy an interlacingcondition, proving a conjecture of McConville, Smyth and Sagan, which in turnimplies a previous conjecture of Morier-Genoud and Ovsienko that r(q;beta) isunimodal. We show that, when beta has an odd number of parts, then thepolynomial is also partially symmetric: the number of ideals of F(beta) of sizek equals the number of filters of size k, when k is below a certain value. Ourproof is completely bijective. Oguz and Ravichandran also introduced a circularversion of fences and proved, using algebraic techniques, that the distributivelattice for such a poset is rank symmetric. We give a bijective proof of thisresult as well. We end with some questions and conjectures raised by this work.

Partial rank symmetry of distributive lattices for fences