The Whitney duals of a graded poset

Abstract

We introduce the notion of a Whitney dual of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define new types of edge labelings which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual. Moreover, we show how to explicitly construct a Whitney dual using a technique involving quotient posets.As applications of our main theorem, we show that geometric lattices and the lattice of noncrossing partitions all have Whitney duals. Our technique gives a combinatorial description of the Whitney dual of the partition lattice in terms of a poset of increasing forests. More generally we give combinatorial descriptions of Whitney duals of geometric lattices in terms of NBC sets. We also provide a combinatorial description of a Whitney dual of the noncrossing partition lattice in terms of collections of decorated Dyck paths.Finally, we show that a graded poset with a Whitney labeling admits a local action of the 0-Hecke algebra of type A on its set of maximal chains. The characteristic of the associated representation is Ehrenborg's flag quasisymmetric function of the poset. The existence of this action implies, using a result of McNamara, that when the maximal intervals of the constructed Whitney duals are bowtie-free, they are also snellable. In the case where these maximal intervals are lattices, they are supersolvable.