Padlock Solitaire: A martingale trick for combinatorial enumeration

TL;DR

This paper introduces Padlock Solitaire, a solitaire game used as a tool to provide simple proofs for classical combinatorial results such as ballot theorems and spanning tree enumeration.

Contribution

It presents a novel game-based approach to combinatorial proofs, linking game analysis with classical enumeration problems.

Findings

Provides simple proofs of ballot theorems

Enumerates spanning trees in graphs and hypergraphs

Introduces Padlock Solitaire as a combinatorial tool

Abstract

We introduce a one-person game that we call Padlock Solitaire which resembles the well-known clock solitaire card game. Analyzing variants of this game we obtain simple proofs of some classical results of combinatorics including ballot theorems and the enumeration of spanning trees in various graphs and hypergraphs.