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

TL;DR

This paper explores the enumerative properties of a class of no-strategy combinatorial games modeled as graded posets, revealing new sequences and interpretations related to classic combinatorial structures.

Contribution

It introduces a novel poset model for certain strategy games and derives new enumerative sequences, connecting them to well-known combinatorial triangles.

Findings

Derived new sequences related to game end states.

Connected enumerative properties to Catalan and Motzkin triangles.

Provided combinatorial interpretations for these sequences.

Abstract

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.