c -Birkhoff polytopes

In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan’s question, in this paper we define a pattern-avoiding Birkhoff polytope called a c -Birkhoff polytope for each Coxeter element c of the symmetric group. We then show that the c -Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the c -sorting word of the longest permutation. When c = s 1 s 2 ⋯ s n , this result recovers an affirmative answer to Davis and Sagan’s question. Another consequence of this result is that the normalized volume of the c -Birkhoff polytope is the number of the longest chains in the (type A) c -Cambrian lattice.