We prove that the multiplication map *W*/*V* x *V* -> *W* for a
generalized quotient of the symmetric group is always surjective when
*V* is an order ideal in right weak order; interpreting these sets of
permutations as linear extensions of 2-dimensional posets gives the
first direct combinatorial proof of an inequality due originally to
Sidorenko in 1991, answering an open problem Morales, Pak, and Panova.
We show that this multiplication map is a bijection if and only if *V*
is an order ideal in right weak order generated by a separable
element, thereby classifying those generalized quotients which induce
*splittings* of the symmetric group, answering a question of
Björner and Wachs (1988). All of these results are conjectured to
extend to arbitrary finite Weyl groups.

Next, we show that separable elements in *W* are in bijection with the
faces of all dimensions of two copies of the graph associahedron of
the Dynkin diagram of *W*. This correspondence associates to each
separable element *w* a certain *nested set*; we give elegant
product formulas for the rank generating functions of the principal
upper and lower order ideals generated by *w* in terms of these nested
sets.

Finally we show that separable elements, although initially defined
recursively, have a non-recursive characterization in terms of root
system pattern avoidance in the sense of Billey and Postnikov.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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