World Cup Enumerative Combinatorics

There are (r+1)(r+2)(2r+3)(r²+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 score-board, and generates World-Cup 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 round-robin 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 line-sums 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