A refinement of weak order intervals into distributive lattices

Details A refinement of weak order intervals into distributive lattices A refinement of weak order intervals into distributive lattices

2011-02-14

arxiv.org/abs/1102.2689v2

Mathematics

Combinatorics

18 pages, 2 figures. The update contains an additional figure, shorter exposition regarding identifying the join-irreducibles,

Scientific paper

In this paper we consider arbitrary intervals in the left weak order of the symmetric group. We show that the Lehmer codes of permutations in an interval forms a distributive lattice under the product order. Furthermore, the rank-generating function of this distributive lattice matches that of the weak order interval. We construct a poset such that the order ideals of the poset, ordered by inclusion, is isomorphic to the poset of Lehmer codes of permutations in the interval. We show that there are at least $(\lfloor\frac{n}{2}\rfloor)!$ permutations in $S_n$ that form a rank-symmetric interval in the weak order.

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