There are $(r+1)(r+2)(2r+3)(r^2+3r+5)$ Ways For the Four Teams of a World Cup Group to Each Have $r$ Goals For and $r$ Goals Against [Thanks to the Soccer Analog of Prop. 4.6.19 of Richard Stanley's (Classic!) EC1]

TL;DR

This paper applies an algebraic combinatorics theorem, originally about magic squares, to enumerate possible outcomes in a World Cup group where four teams have equal goals for and against, revealing a surprising connection.

Contribution

It introduces a novel application of Richard Stanley's theorem to count World Cup group outcomes with equal goals for and against.

Findings

Exact enumeration formula for the outcomes

Connection between combinatorics and sports outcomes

Surprising application of algebraic combinatorics

Abstract

This short tribute to the guru of Enumerative and Algebraic Combinatorics started out when one the authors(DZ) attended the Stanely@70 conference, that took place at the same time as the preliminary stage of the 2014 World Cup. It states a surprising application of an analog of Richard Stanley's famous theorem about the enumeration of magic squares to the enumeration of possible outcomes in a World Cup Group.