# Gray codes

This is based on work done about 20 years ago with a former student Jim McShea.

Gray codes were introduced by Bell Labs physicist Frank Gray in the 1950s. As introduced, a Gray code is an ordering of the n-tuples in $GF(2)^n = \{0,1\}^n$ such that two successive terms differ in only one position. A Gray code can be regarded as a Hamiltonian path in the cube graph. For example:

[[0, 0, 0], [1, 0, 0], [1, 1, 0], [0, 1, 0], [0, 1, 1], [1, 1, 1], [1, 0, 1], [0, 0, 1]]

These can be generalized to n-tuples of integers (mod m) in the obvious way.

Gray codes have several applications:

• solving puzzles such as the Tower of Hanoi and the Brain [G],
• analog-digital-converters (goniometers) [S],
• Hamiltonian circuits in hypercubes [Gil] and Cayley graphs of Coxeter groups [CSW],
• capanology (the study of bell-ringing) [W],
• continuous space-filling curves [Gi],
• classification of Venn diagrams [R],
• design of communication codes,
• and more (see Wikipedia). The Brain puzzle

Here's a SageMath/Python function for computing Gray codes.
def graycode(length,modulus):
"""
Returns the n-tuple reverse Gray code mod m.

EXAMPLES:
sage: graycode(2,4)

[[0, 0],
[1, 0],
[2, 0],
[3, 0],
[3, 1],
[2, 1],
[1, 1],
[0, 1],
[0, 2],
[1, 2],
[2, 2],
[3, 2],
[3, 3],
[2, 3],
[1, 3],
[0, 3]]

"""
n,m = length,modulus
F = range(m)
if n == 1:
return [[i] for i in F]
L = graycode(n-1, m)
M = []
for j in F:
M = M+[ll+[j] for ll in L]
k = len(M)
Mr = *m
for i in range(m-1):
i1 = i*int(k/m)
i2 = (i+1)*int(k/m)
Mr[i] = M[i1:i2]
Mr[m-1] = M[(m-1)*int(k/m):]
for i in range(m):
if is_odd(i):
Mr[i].reverse()
M0 = []
for i in range(m):
M0 = M0+Mr[i]
return M0



REFERENCES

[CSW] J. Conway, N. Sloane, and A. Wilks, “Gray codes and reflection groups”, Graphs and combinatorics 5(1989)315-325

[E] M. C. Er, “On generating the N-ary reflected Gray codes”, IEEE transactions on computers, 33(1984)739-741

[G] M. Gardner, “The binary Gray code”, in Knotted donuts and other mathematical entertainments, F. H. Freeman and Co., NY, 1986

[Gi] W. Gilbert, “A cube-filling Hilbert curve”, Math Intell 6 (1984)78

[Gil] E. Gilbert, “Gray codes and paths on the n-cube”, Bell System Technical Journal 37 (1958)815-826

[R] F. Ruskey, “A Survey of Venn Diagrams“, Elec. J. of Comb.(1997), and updated versions.

[W] A. White, “Ringing the cosets”, Amer. Math. Monthly 94(1987)721-746

# Sports ranking methods, 3

This is the third of a series of expository posts on matrix-theoretic sports ranking methods. This post discusses the random walker ranking.

We follow the presentation in the paper by Govan and Meyer (Ranking National Football League teams using Google’s PageRank). The table of “score differentials” based on the table in a previous post is: $\begin{tabular}{c|cccccc} \verb+x\y+ & Army & Bucknell & Holy Cross & Lafayette & Lehigh & Navy \\ \hline Army & 0 & 0 & 1 & 0 & 0 & 0 \\ Bucknell & 2 & 0 & 0 & 2 & 3 & 0 \\ Holy Cross & 0 & 3 & 0 & 4 & 14 & 0 \\ Lafayette & 10 & 0 & 0 & 0 & 0 & 0 \\ Lehigh & 2 & 0 & 0 & 11 & 0 & 0 \\ Navy & 11 & 14 & 8 & 22 & 6 & 0 \\ \end{tabular}$
This leads to the following matrix: $M_0=\left(\begin{array}{cccccc} 0 & 0 & 1 & 0 & 0 & 0 \\ 2 & 0 & 0 & 2 & 3 & 0 \\ 0 & 3 & 0 & 4 & 14 & 0 \\ 10 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 11 & 0 & 0 \\ 11 & 14 & 8 & 22 & 6 & 0 \\ \end{array}\right) .$

The edge-weighted score-differential graph associated to $M_0$ (regarded as a weighted adjacency matrix) is in the figure below. This matrix $M_0$ must be normalized to create a (row) stochastic matrix: $M = \left(\begin{array}{cccccc} 0 & 0 & 1 & 0 & 0 & 0 \\ {2}/{7} & 0 & 0 /{7} /{7} & 0 \\ 0 /{7} & 0 /{21} /{3} & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ {2}/{13} & 0 & 0 /{13} & 0 & 0 \\ {11}/{61} /{61} /{61} /{61} /{61} & 0 \\ \end{array}\right) .$

Next, to insure it is irreducible, we replace $M$ by $A=(M+J)/2$, where $J$ is the $6\times 6$ doubly stochastic matrix with every entry equal to $1/6$: $A=\left(\begin{array}{cccccc} {1}/{12} & 1/{12} & 7/{12} & 1/{12} & 1/{12} & 1/{12} \\ {19}/{84} & 1/{12} & 1/{12} & 19/{84} & 25/{84} & 1/{12} \\ {1}/{12} & 13/{84} & 1/{12} & 5/{28} & 5/{12} & 1/{12} \\ {7}/{12} & 1/{12} & 1/{12} & 1/{12} & 1/{12} & 1/{12} \\ {25}/{156} & 1/{12} & 1/{12} & 79/{156} & 1/{12} & 1/{12} \\ {127}/{732} & 145/{732} & 109/{732} & 193/{732} & 97/{732} & 1/{12} \end{array}\right).$

Let ${\bf v}_0 = \left( \frac{1}{6} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6} , \frac{1}{6}\right).$

The ranking determined by the random walker method is the reverse of the left eigenvector of $A$ associated to the largest eigenvalue $\lambda_{max}=1$ (by reverse, I mean that the vector ranks the teams from worst-to-best, not from best-to-worst, as we have seen in previous ranking methods).
In other words, the vector ${\bf r}^*=\lim_{n\to \infty}{\bf v}_0A^n.$

This is approximately ${\bf r}^* \cong \left(0.2237\dots ,\,0.1072\dots ,\,0.2006\dots ,\,0.2077\dots ,\,0.1772\dots ,\,0.0833\dots \right).$

Its reverse gives the ranking:

Army $<$ Lafayette $<$ Bucknell $<$ Lehigh $<$ Holy Cross $<$ Navy.

This gives a prediction failure rate of $13.3\%$.

# Simple unsolved math problem, 6

If you know a little point-set topology, below is an unsolved math problem whose statement is relatively simple.

Probably everyone has at least seen the Mandelbrot set in some form, as it’s a popular object of mathematical artists. Here’s a picture from Wikipedia: The formal definition is as follows. Let $f_c (z)=z^2+c$, where $c\in \mathbb{C}$ is a complex number. The Mandelbrot set $X$ is the complex plot of the set of complex numbers $c$ for which the sequence of iterates $f_c (0), f_c (f_c (0)), f_c (f_c (f_c (0))), \dots,$

remains bounded in absolute value.
We say $X$ is locally connected if every point $x\in X$ admits a neighborhood basis consisting entirely of open, connected sets.

Conjecture: The Mandelbrot set $X$ is locally connected.

# A tribute to TS Michael

I’ve known TS for over 20 years as a principled colleague and a great teacher. TS at the USNA in Dec 2015.

However, we really never spoke much except for the past five-to-ten years or so. For a period, I wrote a lot about error-correcting codes and we’d talk occasionally about our common interests (for example, I found his paper “The rigidity theorems of Hamada and Ohmori, revisited” fascinating). However, once I became interested in graph theory, we spoke as often as I could corner him. He taught me a lot and only know I realize how lucky I was to have him as a colleague.

I remember many times, late on a Friday, when we’d talk for an hour or two about chess, mathematics, “office politics” (he always knew more than me), and allergies. Here’s one of his favorite chess problems: Mate in 549 moves. This problem was discovered by a team of chess engame experts at Lomonosov University, Moscow, August 2012.

Maybe this says more about me than him, but when it was just the two of us, we rarely talked about families or relationships. None-the-less, he always treated me like a good friend. One of my favorite memories was when my wife and I were shopping at the plaza where his condo building was located (it’s a big plaza). Elva and I were walking store-to-store when we spotted TS. He was walking to distract himself from his discomfort. At the time, doctors didn’t know what his problems were and he suspected allergies. I have a number of food sensitivities and he was a welcomed fountain of medical knowledge about these issues. (In fact, his hints have really helped me a lot, health-wise.) In any case, TS and Elva and I spoke for 30 minutes or so about health and family. I remember how gracious and thoughtful he was, skillfully steering the conversation into non-technical matters for Elva’s benefit. I ran into him another time while waiting for Elva, who was in a nearby doctor’s office (I told you this was a big shopping plaza). TS generously waited with me until Elva was ready to be picked up. What we chatted about is lost in the cobwebs of my memory but I remember vividly where we sat and the kind of day it was. TS had such a kind heart.

As I said, TS taught me a lot about graph theory. Whether in-between classes or when I was lucky enough to spot him late in the day, he’d kindly entertain my naive (usually false) conjectures and speculations about strongly regular graphs. I never heard him speak in anything but the kindest terms. He’d never say “that’s just plain wrong” or “idiotic” (even if it was) but instead teach me the correct way to think about it in a matter in which I could see myself how my speculations were wrong-headed. My upcoming book with Caroline Melles is indebted to his insight and suggestions.

Even after he left Maryland to spend his remaining days with his family in California, TS emailed encouragement and suggestions about an expository paper I was writing to help connect my matrix theory students with the methods of ranking sports teams. While he was very helpful and provided me with his excellent insights as usual, in truth, I used the work on the paper as an excuse to keep up with his health status. I’m relatively ignorant of medical issues and tried to stay optimistic until it’s totally unrealistic. As sad as it was, we was always frank and honest with me about his prognosis.

He’s gone now, but as a teacher, researcher, and as a kind soul, TS is unforgettable.

A list of TS’s publications:

1. T. S. Michael, Tournaments, book chapter in Handbook of Linear Algebra, 2nd ed, CRC Press, Boca Raton, 2013.
2. T. S. Michael, Cycles of length 5 in triangle-free graphs: a sporadic counterexample to a characterization of equality, Bulletin of the Institute of Combinatorics and Its Applications, 67 (2013) 6–8.
3. T. S. Michael and Val Pinciu, Guarding orthogonal prison yards: an upper bound,
Congressus Numerantium, 211 (2012) 57–64.
4. Ilhan Hacioglu and T. S. Michael, The p-ranks of residual and derived skew Hadamard designs,
Discrete Mathematics, 311 (2011) 2216-2219.
5. T. S. Michael, Guards, galleries, fortresses, and the octoplex, College Math Journal, 42 (2011) 191-200. (This paper won a Polya Award)
6. Elizabeth Doering, T. S. Michael, and Bryan Shader, Even and odd tournament matrices with minimum rank over finite fields, Electronic Journal of Linear Algebra, 22 (2011) 363-377.
7. Brenda Johnson, Mark E. Kidwell, and T. S. Michael, Intrinsically knotted graphs have at least 21 edges, Journal of Knot Theory and Its Ramifications, 19 (2010) 1423-1429.
8. T. S. Michael, How to Guard an Art Gallery and Other Discrete Mathematical Adventures. Johns Hopkins University Press, Baltimore, 2009.
9. T. S. Michael and Val Pinciu, Art gallery theorems and triangulations, DIMACS Educational Module Series, 2007, 18 pp (electronic 07-1)
10. T. S. Michael and Thomas Quint, Sphericity, cubicity, and edge clique covers of graphs, Discrete Applied Mathematics, 154 (2006) 1309-1313.
11. T. S. Michael and Val Pinciu, Guarding the guards in art galleries, Math Horizons, 14 (2006), 22-23, 25.
12. Richard J. Bower and T. S. Michael, Packing boxes with bricks, Mathematics Magazine, 79 (2006), 14-30.
13. T. S. Michael and Thomas Quint, Optimal strategies for node selection games: skew matrices and symmetric games, Linear Algebra and Its Applications 412 (2006) 77-92.
14. T. S. Michael, Ryser’s embedding problem for Hadamard matrices, Journal of Combinatorial Designs 14 (2006) 41-51.
15. Richard J. Bower and T. S. Michael, When can you tile a box with translates of two given rectangular bricks?, Electronic Journal of Combinatorics 11 (2004) Note 7, 9 pages.
16. T. S. Michael and Val Pinciu, Art gallery theorems for guarded guards, Computational Geometry 26 (2003) 247-258.
17. T. S. Michael, Impossible decompositions of complete graphs into three Petersen subgraphs, Bulletin of the Institute of Combinatorics and Its Applications 39 (2003) 64-66.
18. T. S. Michael and William N. Traves, Independence sequences of well-covered graphs: non-unimodality and the roller-coaster conjecture, Graphs and Combinatorics 19 (2003) 403-411.
19. T. S. Michael and Thomas Quint, Sphere of influence graphs and the L-Infinity metric, Discrete Applied Mathematics 127 (2003) 447-460.
20. T. S. Michael, Signed degree sequences and multigraphs, Journal of Graph Theory 41 (2002) 101-105.
21. T. S. Michael and Val Pinciu, Multiply guarded guards in orthogonal art galleries, Lecture Notes in Computer Science 2073, pp 753-762, in: Proceedings of the International Conference on Computer Science, San Francisco, Springer, 2001.
22. T. S. Michael, The rigidity theorems of Hamada and Ohmori, revisited, in Coding Theory and Cryptography: From the Geheimschreiber and Enigma to Quantum Theory. (Annapolis, MD, 1998), 175-179, Springer, Berlin, 2000.
23. T. S. Michael and Thomas Quint, Sphere of influence graphs in general metric spaces, Mathematical and Computer Modelling, 29 (1999) 45-53.
24. Suk-Geun Hwang, Arnold R. Kraeuter, and T. S. Michael, An upper bound for the permanent of a nonnegative matrix, Linear Algebra and Its Applications 281 (1998), 259-263.
* First Corrections: Linear Algebra and Its Applications 300 (1999), no. 1-3, 1-2
25. T. S. Michael and W. D. Wallis, Skew-Hadamard matrices and the Smith normal form, Designs, Codes, and Cryptography, 13 (1998) 173-176.
26. T. S. Michael, The p-ranks of skew Hadamard designs, Journal of Combinatorial Theory, Series A, 73 (1996) 170-171.
27. T. S. Michael, The ranks of tournament matrices, American Mathematical Monthly, 102 (1995) 637-639.
28. T. S. Michael, Lower bounds for graph domination by degrees, pp 789-800 in Graph Theory, Combinatorics, and Algorithms: Proceedings of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs, Y. Alavi and A. Schwenk (eds.), Wiley, New York, 1995.
29. T. S. Michael and Thomas Quint, Sphere of influence graphs: a survey, Congressus Numerantium, 105 (1994) 153-160.
30. T. S. Michael and Thomas Quint, Sphere of influence graphs: edge density and clique size, Mathematical and Computer Modelling, 20 (1994) 19-24.
31. T. S. Michael and Aaron Stucker, Mathematical pitfalls with equivalence classes, PRIMUS, 3 (1993) 331-335.
32. T. S. Michael, The structure matrix of the class of r-multigraphs with a prescribed degree sequence, Linear Algebra and Its Applications, 183 (1993) 155-177.
33. T. S. Michael, The decomposition of the complete graph into three isomorphic strongly regular graphs, Congressus Numerantium, 85 (1991) 177-183.
34. T. S. Michael, The structure matrix and a generalization of Ryser’s maximum term rank formula, Linear Algebra and Its Applications, 145 (1991) 21-31.
35. Richard A. Brualdi and T. S. Michael, The class of matrices of zeros, ones and twos with prescribed row and column sums, Linear Algebra and Its Applications, 114(115) (1989) 181-198.
36. Richard A. Brualdi and T. S. Michael, The class of 2-multigraphs with a prescribed degree sequence, Linear and Multilinear Algebra, 24 (1989) 81-102.
37. Richard A. Brualdi, John L. Goldwasser, and T. S. Michael, Maximum permanents of matrices of zeros and ones, Journal of Combinatorial Theory, Series A, 47 (1988) 207-245.

# Memories of TS Michael, by Bryan Shader

TS Michael passed away on November 22, 2016, from cancer. I will miss him as a colleague and a kind, wise soul. TS Michael in December 2015 at the USNA

Bryan Shader has kindly allowed me to post these reminiscences that he wrote up.

Memories of TS Michael, by Bryan Shader

Indirect influence
TS indirectly influenced my choice of U. Wisconsin-Madison for graduate school. My senior year as an undergraduate, Herb Ryser gave a talk at my school. After the talk I was able to meet Ryser and asked for advice on graduate schools. Herb indicated that one of his very good undergraduate students had chosen UW-Madison and really liked the program. I later found out that the person was TS.

Back in the dark ages, universities still did registration by hand. This meant that for a couple of days before each semester the masses of students would wind their way through a maze of stations in a large gymnasium. For TS’s first 4 years, he would invariably encounter a road block because someone had permuted the words in his name (Todd Scott Michael) on one of the forms. After concretely verifying the hatcheck probabilities and fearing that this would cause some difficulties in graduating, he legally changed his name to TS Michael.

Polyominoes & Permanents
I recall many stories about how TS’s undergraduate work on polyominoes affected
his life. In particular, he recalled how once he started working on tilings on
polyominoes, he could no longer shower, or swim without visualizing polynomino
tilings on the wall’s or floor’s tiling. We shared an interest and passion for permanents (the permanent is a function of a matrix much like the determinant and plays a critical role in combinatorics). When working together we frequently would find that we both couldn’t calculate the determinant of a 3 by 3 matrix correctly, because we were calculating the permanent rather than the determinant.

Presentations and pipe-dreams
TS and I often talked about how best to give a mathematical lecture, or
presentation at a conference. Perhaps this is not at all surprising, as our common advisor (Richard Brualdi) is an incredible expositor, as was TS’s undergraduate advisor (Herb Ryser, our mathematical grandfather). TS often mentioned how Herb Ryser scripted every moment of a lecture; he knew each word he would write on the board and exactly where it would be written. TS wasn’t quite so prescriptive–but before any presentation he gave he would go to the actual room of the presentation a couple of times and run through the talk. This would include answering questions from the “pretend” audience. After being inspired by TS’s talks, I adopted this preparation method.
TS and I also fantasized about our talks ending with the audience lifting us up on their shoulders and carrying us out of the room in triumph! That is never happened to either of us (that I know of), but to have it, as a dream has always been a good motivation.

Mathematical heritage
TS was very interested in his mathematical heritage, and his mathematics brothers and sisters. TS was the 12th of Brandi’s 37 PhD students; I was the 15th. In 2005, TS and I organized a conference (called the Brualidfest) in honor of Richard Brualdi. Below I attach some photos of the design for the T-shirt. t-shirt design for Brualdi-Fest, 1

The first image shows a biclique partition of K_5; for each color the edges of the given color form a complete bipartite graph; and each each of the completed graph on 5 vertices is in exactly one of these complete bipartite graph. This is related to one of TS’s favorite theorem: the Graham-Pollak Theorem. t-shirt design for Bruldi-Fest, 2

The second image (when the symbols are replaced by 1s) is the incidence matrix of the projective plane of order 2; one of TS’s favorite matrices.

Here’s a photo of the Brualdi and his students at the conference: From L to R they are: John Mason (?), Thomas Forreger, John Goldwasser, Dan Pritikin, Suk-geun Hwang, Han Cho, T.S. Michael, B. Shader, Keith Chavey, Jennifer Quinn, Mark Lawrence, Susan Hollingsworth, Nancy Neudauer, Adam Berliner, and Louis Deaett.

Here’s a picture for a 2012 conference: From bottom to top: T.S. Michael (1988), US Naval Academy, MD; Bryan Shader (1990), University of Wyoming, WY; Jennifer Quinn (1993), University of Washington, Tacoma, WA; Nancy Neudauer (1998), Pacific University, OR; Susan Hollingsworth (2006), Edgewood College, WI; Adam Berliner (2009), St. Olaf College, MN; Louis Deaett (2009), Quinnipiac University, CT; Michael Schroeder (2011), Marshall University, WV; Seth Meyer (2012), Kathleen Kiernan (2012).

Here’s a caricature of TS made by Kathy Wilson (spouse of mathematician
Richard Wilson) at the Brualdifest: TS Michael, by Kathy Wilson

Long Mathematical Discussions
During graduate school, TS and I would regularly bump into each other as we
were coming and going from the office. Often this happened as we were crossing University Avenue, one of the busiest streets in Madison. The typical conversation started with a “Hi, how are you doing? Have you considered X?” We would then spend the next 60-90 minutes on the street corner (whether it was a sweltering 90 degrees+, or a cold, windy day) considering X. In more recent years, these conversations have moved to hotel lobbies at conferences that we attend together. These discussions have been some of the best moments of my life, and through them I have become a better mathematician.

Here’s a photo of T.S. Michael with Kevin van der Meulen at the Brualdi-fest. I’m guessing they are in the midst of one of those “Have you considered X?” moments that TS is famous for.

Mathematical insight
TS has taught me a lot about mathematics, including:

•  How trying to generalize a result can lead to better understanding of the original result.
•  How phrasing a question appropriately is often the key to a mathematical breakthrough
• Results that are surprising (e.g. go against ones intuition), use an elegant proof (e.g. bring in matrices in an unexpected way), and are aesthetically pleasing are worth pursing (as Piet Hein said “Problems worthy of attack, prove their worth by fighting back.”)
•  The struggle to present the proof of a result in the simplest, most self-contained way is important because often it produces a better understanding. If you can’t say something in a clean way, then perhaps you really don’t understand it fully.

TS’ mathematics fathers are:
Richard Brualdi ← Herb Ryser ← Cyrus MacDuffee ← Leonard Dickson ← E.H. Moore ← H. A. Newton ← Michel Chasles ← Simeon Poisoon ← Joseph Lagrange ← Leonhard Euler ← Johann Bernoulli.

# Simple unsolved math problem, 5

This is now almost completely solved! Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Joni Teräväinen, and Terrance Tao solved the conjecture below in the “interior” of Pascal’s triangle (see T. Tao’s blog post for further details, with the link to the paper and a discussion).

It seems everyone’s heard of Pascal’s triangle. However, if you haven’t then it is an infinite triangle of integers with 1‘s down each side and the inside numbers determined by adding the two numbers above it: First 6 rows of Pascal’s triangle

The first 6 rows are depicted above. It turns out, these entries are the binomial coefficients that appear when you expand $(x+y)^n$ and group the terms into like powers $x^{n-k}y^k$: First 6 rows of Pascal’s triangle, as binomial coefficients.

The history of Pascal’s triangle pre-dates Pascal, a French mathematician from the 1600s, and was known to scholars in ancient Persia, China, and India.

Starting in the mid-to-late 1970s, British mathematician David Singmaster was known for his research on the mathematics of the Rubik’s cube. However, in the early 1970’s, Singmaster made the following conjecture .

Conjecture: If $N(a)$ denotes the number of times the number $a > 1$ appears in Pascal’s triangle then $N(a) \leq 12$ for all $a>1$.

In fact, there are no known numbers $a>1$ with $N(a)>8$ and the only number greater than one with $N(a)=8$ is a=3003.

References:

 Singmaster, D. “Research Problems: How often does an integer occur as a binomial coefficient?”, American Mathematical Monthly, 78(1971) 385–386.

# Simple unsolved math problem, 4

Problem: Optimally pack n unit circles into the smallest possible equilateral triangle. Let L(n) denote the length of the side of the smallest equilateral triangle in which n circles have been packed optimally. This number is, in general, unknown.

# The minimum upset ranking problem

Suppose n teams play each other, and let Team $r_1 <$ Team $r_2 < \dots <$ Team $r_n$ denote some fixed ranking (where $r_1,\dots,r_n$ is some permutation of $1,\dots,n$). An upset occurs when a lower ranked team beats an upper ranked team. For each ranking, ${\bf r}$, let $U({\bf r})$ denote the total number of upsets. The minimum upset problem is to find an “efficient” construction of a ranking for which $U({\bf r})$ is as small as possible.

In general, let $A_{ij}$ denote the number of times Team i beat team $j$ minus the number of times Team j beat Team i. We regard this matrix as the signed adjacency matrix of a digraph $\Gamma$. Our goal is to find a Hamiltonian (undirected) path through the vertices of $\Gamma$ which goes the “wrong way” on as few edges as possible.

1. Construct the list of spanning trees of $\Gamma$ (regarded as an undirected graph).
2. Construct the sublist of Hamiltonian paths (from the spanning trees of maximum degree 2).
3. For each Hamiltonian path, compute the associated upset number: the total number of edges transversal in $\Gamma$ going the “right way” minus the total number going the “wrong way.”
4. Locate a Hamiltonian for which this upset number is as large as possible.

Use this sagemath/python code to compute such a Hamiltonian path.

def hamiltonian_paths(Gamma, signed_adjacency_matrix = []):
"""
Returns a list of hamiltonian paths (spanning trees of
max degree <=2).

EXAMPLES:
sage: Gamma = graphs.GridGraph([3,3])
sage: HP = hamiltonian_paths(Gamma)
sage: len(HP)
20
sage: A = matrix(QQ,[
[0 , -1 , 1  , -1 , -1 , -1 ],
[1,   0 ,  -1,  1,  1,   -1  ],
[-1 , 1 ,  0 ,  1 , 1  , -1  ],
[1 , -1 , -1,  0 ,  -1 , -1  ],
[1 , - 1 , - 1 , 1 , 0 , - 1  ],
[1 ,  1  ,  1  , 1  , 1  , 0 ]
])
sage: HP = hamiltonian_paths(Gamma, signed_adjacency_matrix = A)
sage: L = [sum(x) for x in HP]; max(L)
5
sage: L.index(5)
21
sage: HP
[Graph on 6 vertices,
[0, 5, 2, 1, 3, 4],
[-1, 1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1]]
sage: L.count(5)
1

"""
ST = Gamma.spanning_trees()
HP = []
for X in ST:
L = X.degree_sequence()
if max(L)<=2:
#print L,ST.index(X), max(L)
HP.append(X)
return HP
HP = []
for X in ST:
L = X.degree_sequence()
if max(L)<=2:
#VX = X.vertices()
EX = X.edges()
if EX != EX[-1]:
ranking = X.shortest_path(EX,EX[-1])
else:
ranking = X.shortest_path(EX,EX[-1])
signature = [A[ranking[i]][ranking[j]] for i in range(len(ranking)-1) for j in range(i+1,len(ranking))]
HP.append([X,ranking,signature])
return HP



Wessell describes this method in a different way.

1. Construct a matrix, $M=(M_{ij})$, with rows and columns indexed by the teams in some fixed order. The entry in the i-th row and the j-th column is defined by $m_{ij}= \left\{ \begin{array}{rr} 0,& {\rm if\ team\ } i {\rm \ lost\ to\ team\ } j,\\ 1,& {\rm if\ team\ } i {\rm\ beat\ team\ } j,\\ 0, & {\rm if}\ i=j. \end{array} \right.$
2. Reorder the rows (and corresponding columns) to in a basic win-loss order: the teams that won the most games go at the
top of $M$, and those that lost the most at the bottom.
3. Randomly swap rows and their associated columns, each time checking if the
number of upsets has gone down or not from the previous time. If it has gone down, we keep
the swap that just happened, if not we switch the two rows and columns back and try again.

An implementaiton of this in Sagemath/python code is:

def minimum_upset_random(M,N=10):
"""
EXAMPLES:
sage: M = matrix(QQ,[
[0 , 0 , 1  , 0 , 0 , 0 ],
[1,   0 ,  0,  1,  1,   0  ],
[0 , 1 ,  0 ,  1 , 1  , 0  ],
[1 , 0 , 0,  0 ,  0 , 0  ],
[1 , 0 , 0 , 1 , 0 , 0  ],
[1 ,  1  ,  1  , 1  , 1  , 0 ]
])
sage: minimum_upset_random(M)
(
[0 0 1 1 0 1]
[1 0 0 1 0 1]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[1 1 1 1 0 1]
[0 0 1 1 0 0], [1, 2, 0, 3, 5, 4]
)

"""
n = len(M.rows())
Sn = SymmetricGroup(n)
M1 = M
wins = sum([sum([M1[j][i] for i in range(j,6)]) for j in range(6)])
g0 = Sn(1)
for k in range(N):
g = Sn.random_element()
P = g.matrix()
M0 = P*M1*P^(-1)
if sum([sum([M0[j][i] for i in range(j,6)]) for j in range(6)])>wins:
M1 = M0
g0 = g*g0
return M1,g0(range(n))


# Sage and the future of mathematics

I am not a biologist nor a chemist, and maybe neither are you, but suppose we were. Now, if I described a procedure, in “standard” detail, to produce a result XYZ, you would (based on your reasonably expected expertise in the field), follow the steps you believe were described and either reproduce XYZ or, if I was mistaken, not be able to reproduce XYZ. This is called scientific reproducibility. It is cructial to what I believe is one of the fundamental principles of science, namely Popper’s Falsifiability Criterion.

More and more people are arguing, correctly in my opinion, that in the computational realm, in particular in mathematical research which uses computational experiments, that much higher standards are needed. The Ars Technica article linked above suggests that “it’s time for a major revision of the scientific method.” Also, Victoria Stodden argues one must “facilitate reproducibility. Without this data may be open, but will remain de facto in the ivory tower.” The argument basically is that to reproduce computational mathematical experiments is unreasonably difficult, requiring more that a reasonable expertise. These days, it may in fact (unfortunately) require purchasing very expensive software, or possessing of very sophisticated programming skills, accessibility to special hardware, or (worse) guessing parameters and programming procedures only hinted at by the researcher.

Hopefully, Sage can play the role of a standard bearer for such computational reproducibility. Sage is free, open source and there is a publically available server it runs on (sagenb.org).

What government agencies should require such reproducibility? In my opinion, all scientific funding agencies (NSF, etc) should follow these higher standards of computational accountability.