Enumerative Properties of Posets Corresponding to a Certain Class of No Strategy Games

In this paper, we consider a game beginning with a multiset of elements from a group. On a move, two elements are replaced by their sum. This is a no strategy game, and can be modeled as a graded poset with the rank of a node equal to the cardinality of its multiset. We study the enumerative properties of certain variations of this game, such as the number of ways to play them and their numbers of end states. This leads to several new sequences, as well as new interpretations of classic sequences such as those found in the Catalan and Motzkin triangles.