Posets in which every interval is a product of chains, and natural local actions of the symmetric group Original Research Article

Consider a graded poset P with maximal and minimal elements. If everey interval of rank three in P is a product of chains, and for everey interval [x, y] of rank at least four, the open interval (x, y) is connected, we show that the entire poset is a product of chains. This proves a conjecture of Stanley concerning the natural local action of the symmetric group on maximal chains in a graded poset. On a product of chains, the natural action is thepermutation action on the multiset corresponding to the chains. The result implies that if every open interval is connected (which holds if the poset is Cohen-Macaulay, for example), the only possible such natural action is this multiset action.