  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.
