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]
By
Shalosh B. Ekhad and Doron Zeilberger
.pdf
.ps
.tex
First Written: July 7, 2014
[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org]
Dedicated to Richard Stanley (b. June 23, 1944), on his "number of ways for a simple Drunkard to return home after 8 steps"th birthday
This short tribute to the guru of Enumerative and Algebraic Combinatorics started out when
one of us (DZ) attended the Stanley@70 conference,
that took place at the same time as the preliminary stage of the 2014 World Cup.
Maple Packages

GOALS,
uses the polynomial ansatz to discover (rigorously!) polynomial expressions to
lots of enumeration questions related to the Group stage of the World Cup

WorldCup,
figures out how to reconstruct the possible individual games' scores from
the total scoreboard, and generates WorldCup puzzle books.
Some Input and Output files for the Maple package GOALS

If you want to see polynomial expressions, in r, for the number of ways where n teams (for the World Cup take n=4)
can play a roundrobin tournament, where each team scored exactly r Goals For (GF) and r Goals Against (GA), in other
words the number of n by n magic squares with linesums r (famously treated in EC1, Prop. 4.6.19), BUT
with the diagonal entries all 0 (teams do not play with themselves!), for n from 3 to 6,
the input yields
the output

If you want to see
many more polynomial expressions, in r, for the number of ways where 4 teams
can have Goals For, and Goals Against vectors
(r+a1,r+a2,r+a3,r+a4) (r+b1,r+b2,r+b3,r+b4)
For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥0
and b1 ≥b2 ≥b3 ≥b4 ≥0
and a1+a2+a3+a4=b1+b2+b3+b4 ≤ 6
the input yields
the output

If you want to see
many more polynomial expressions, in r, for the number of ways where 4 teams
can have Goals For, and Goals Against vectors
(r+a1,r+a2,r+a3,r+a4) (r+b4,r+b3,r+b2,r+b1)
For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥0
and b1 ≥b2 ≥b3 ≥b4 ≥0
and a1+a2+a3+a4=b1+b2+b3+b4 ≤ 6
the input yields
the output

If you want to see
many more polynomial expressions, in r, for the number of ways where FIVE teams
can have Goals For, and Goals Against vectors
(r+a1,r+a2,r+a3,r+a4, r+a5) (r+b1,r+b2,r+b3,r+b4, r+b5)
For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥ a5 ≥ 0
and b1 ≥b2 ≥b3 ≥b4 ≥ b5 ≥ 0
and a1+a2+a3+a4 +a5 =b1+b2+b3+b4+b5 ≤ 4
the input yields
the output

If you want to see
many more polynomial expressions, in r, for the number of ways where FIVE teams
can have Goals For, and Goals Against vectors
(r+a1,r+a2,r+a3,r+a4, r+a5) (r+b5,r+b4,r+b3,r+b2, r+b1)
For ALL choices of a1 ≥a2 ≥a3 ≥a4 ≥ a5 ≥ 0
and b1 ≥b2 ≥b3 ≥b4 ≥ b5 ≥ 0
and a1+a2+a3+a4 +a5 =b1+b2+b3+b4+b5 ≤ 4
the input yields
the output
Some Input and Output files for the Maple package WorldCup

If you want to see a Soccer puzzle book with more than seventy entertaining puzzles
the input yields
the output

If you want to see a Soccer puzzle book with more than twenty, somewhat more challenging puzzles
the input yields
the output

If you want to see a Soccer puzzle book with twelve yet more challenging puzzles
the input yields
the output

If you want to see a Soccer puzzle book with 22 puzzles but now there are FIVE teams
the input yields
the output
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
Doron Zeilberger's Home Page