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A Refinement of Weak Order Intervals into Distributive Lattices

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Abstract

In this paper we consider arbitrary intervals in the left weak order on the symmetric group S n . We show that the Lehmer codes of permutations in an interval form 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 its lattice of order ideals is isomorphic to the lattice of Lehmer codes of permutations in the given interval. We show that there are at least \({\left(\lfloor {\frac{n}{2}} \rfloor \right)!}\) permutations in S n that form a rank-symmetric interval in the weak order.

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  1. 41 Park Terrace West, APT C1, New York, NY, 10034, USA

    Hugh Denoncourt

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  1. Hugh Denoncourt
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Correspondence to Hugh Denoncourt.

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Denoncourt, H. A Refinement of Weak Order Intervals into Distributive Lattices. Ann. Comb. 17, 655–670 (2013). https://doi.org/10.1007/s00026-013-0200-y

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