Two Generalizations of Posets of Shuffles

We study posets defined by Stanley as a multiset generalization of Greeneʹs posets of shuffles. Ehrenborg defined a quasi-symmetric function encoding for the flag f-vector, denoted FP, and we determine FP for shuffle posets of multisets, expressing it as a Schur-positive symmetric function. This leads to several combinatorial formulas as well as proofs that shuffle posets of multisets are supersolvable and have symmetric chain decompositions. We also generalize posets of shuffles to posets for shuffling k words, answering a question of Stanley. Finally, we extend our results about shuffle posets of multisets to k-shuffle posets.