Bijective proofs for some results on the descent polytope

For a word v in variables x and y, Chebikin and Ehrenborg found that the number of faces of the descent polytope DPv equals the number of factorizations of v using subfactors of the form xy and yx with some additional constraints. They also showed the number of faces of DPv equals the number of alternating subwords of v and raised the problem of finding a bijective proof between these two enumerative results. In this paper, we provide an algorithmically defined combinatorial proof, which also gives a correspondence between factorizations of an xy-word and its reverse. For the alternating descent polytope, we show the faces of the descent polytope are in bijection with certain weighted compositions of n and a class of lattice paths of length n + 1 contained in the region −2 ≤ y ≤ 2.