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In this paper we consider arbitrary intervals in the left weak order on thesymmetric group $S_n$. We show that the Lehmer codes of permutations in aninterval form a distributive lattice under the product order. Furthermore, therank-generating function of this distributive lattice matches that of the weakorder interval. We construct a poset such that its lattice of order ideals isisomorphic to the lattice of Lehmer codes of permutations in the giveninterval. We show that there are at least$\left(\lfloor\frac{n}{2}\rfloor\right)!$ permutations in $S_n$ that form arank-symmetric interval in the weak order.

A refinement of weak order intervals into distributive lattices