This page brought to you by the number n
Prof Ian M. Wanless
A matchbox autobiography
I was born and grew up in Canberra, and studied mathematics at the Australian National University.
My PhD thesis was entitled "Permanents, matchings and Latin
rectangles". It was supervised by
Brendan McKay, who to this day
makes
scurrilous aspersions on my parentage. In 19989 I worked with
Nick Wormald
at the University of Melbourne
on asymptotic enumerations and random graphs. Then in October 1999,
I took up a Junior Research Fellowship at
Christ Church,
Oxford which ended in 2003. Then,
for old times' sake I spent a year or two
back at ANU in Brendan's tender loving care.
In 2005 I had a senior lectureship at
Charles Darwin University, which turned out to be a bad idea because half way through
the year they scrapped all their interesting maths courses.
Since 2006 I have been in the
School of Mathematical Sciences at
Monash University in Melbourne.
In 2011 I was lucky enough to be awarded an Australian Research Council
"Future Fellowship", which I guess means that I am now a future former
future fellow! We have a very active combinatorics group at Monash.
Click
here
to see what we've been up to.
Research Interests
My research is in combinatorics, which seems to be the study of how
many different ways you can define the word `combinatorics'. So let me
be more specific. I'm interested mainly in
Latin squares,
matrix permanents and
graph theory.
If you would like to do a project in one of these areas, check out
some suggested topics.
A Latin square is a matrix in which each number occurs
exactly once in every row and once in every column. For example
here's a Latin square of order 8:
1 5 3 7 2 4 6 8
4 7 5 1 3 6 8 2
3 1 2 4 6 8 5 7
5 3 7 6 8 2 4 1
2 4 6 8 5 1 7 3
7 6 8 2 4 3 1 5
6 8 1 3 7 5 2 4
8 2 4 5 1 7 3 6
Notice that each row and column consists of a permutation of the
set {1,2,3,...,8}. This particular square has some other nice patterns
in it. For example the even numbers occur in cyclic order in each
row, whereas the odd numbers occur in cyclic order in each column.
This square is also unusual in that it has no Latin subsquares.
Here's a research challenge for you: Generalise this pattern to larger
Latin squares. I'm particularly interested in orders that are
powers of two, but I'm genuinely interested in finding this square's
bigger siblings. Email me if you
have any ideas.
Here's another challenge for you. Brendan and I recently counted the
Latin squares of order 11, and it turns out there are rather a lot of
them. 776966836171770144107444346734230682311065600000 to be precise!
If we look at the number of reduced latin squares (meaning their first
row and column are in natural order) then the numbers are as follows:
Order Number of reduced LS
2 1
3 1
4 2^{2}
5 2^{3}.7
6 2^{6}.3.7^{2}
7 2^{10}.3.5.1103
8 2^{17}.3.1361291
9 2^{21}.3^{2}.5231.3824477
10 2^{28}.3^{2}.5.31.37.547135293937
11 2^{35}.3^{4}.5.2801.2206499.62368028479
Question: Why are these numbers divisible by such a high power of two?
From theoretical arguments we can only predict that the last number
here is divisible by 2^{4}, which is a long way short
of 2^{35}!
From time to time I enumerate classes of latin squares so I have a lot
of data sitting on my hard drive. Some of it is available
here,
but if there's something else you want then ask  I might have it.
So what work have I done on Latin squares? Here's a list of my
published or in progress papers in this field (click on a title to
read the abstract).

M. J. L. Mendis and I. M. Wanless,
Autoparatopisms of quasigroups and Latin squares,
submitted (2015).

J. Egan and I. M. Wanless,
Enumeration of MOLS of small order,
Math. Comp., to appear (2015).

D. Donovan, K. Johnson and I. M. Wanless,
Permanents and determinants of Latin squares,
J. Combin. Designs to appear (2015).

M. J. L. Mendis and I. M. Wanless,
Latin squares with a unique intercalate,
J. Combin. Designs to appear (2015).

B. D. McKay, I. M. Wanless and X. Zhang,
The order of automorphisms of quasigroups,
J. Combin. Designs 23 (2015), 275288.

J. Browning, P. J. Cameron and I. M. Wanless,
Bounds on the number of small Latin subsquares,
J. Combin. Theory Ser. A 124 (2014), 4156.

T. Grubman and I. M. Wanless,
Growth rate of canonical and minimal group embeddings
of spherical latin trades,
J. Combin. Theory Ser. A 123 (2014), 5772.

N. J. Cavenagh, J. Kuhl and I. M. Wanless,
Longest partial transversals in plexes,
Ann. Comb. 18 (2014), 419428.

I. M. Wanless and X. Zhang,
On the existence of retransmission permutation arrays,
Disc. Appl. Math., 161 (2013), 27722777.

I. M. Wanless and X. Zhang,
Transversals of latin squares and covering radius of set of permutations,
European J. Combin., 34, (2013) 11301143.

J. Browning, D. S. Stones and I. M. Wanless,
Bounds on the number of autotopisms and subsquares of a Latin square,
Combinatorica, 33, (2013) 1122.

D. Bryant, N. J. Cavenagh, B. Maenhaut, K. Pula and I. M. Wanless,
Nonextendible latin cuboids,
SIAM J. Discrete Math. 26, (2012) 239249.
 J. Barát and I. M. Wanless,
A cube dismantling problem related to bootstrap percolation,
J. Stat. Phys., 149, (2012) 754770.

J. Egan and I. M. Wanless,
Latin squares with restricted transversals,
J. Combin. Designs, 20 (2012), 124141.
(Click
here
for data)

D. S. Stones, P. Vojtechovský and I. M. Wanless,
Cycle structure of autotopisms of quasigroups and Latin squares,
J. Combin. Designs, 20 (2012), 227263.

D. S. Stones and I. M. Wanless,
How not to prove the AlonTarsi conjecture,
Nagoya Math. J. 205 (2012), 124.

D. S. Stones and I. M. Wanless,
A
congruence connecting latin rectangles and partial orthomorphisms,
Ann. Comb., 16 (2012), 349365.

P. Danziger, I. M. Wanless and B. S. Webb,
Monogamous Latin Squares,
J. Combin. Theory Ser. A 118 (2011), 796807.

J. Egan and I. M. Wanless,
Indivisible partitions of latin squares,
J. Statist. Plann. Inference,
141 (2011) 402417.

J. Browning, P. Vojtechovský and I. M. Wanless,
Overlapping latin subsquares and full products,
Comment. Math. Univ. Carolin.,
51 (2010) 175184.

N. J. Cavenagh and I. M. Wanless,
On the number of transversals in Cayley tables of cyclic groups,
Disc. Appl. Math. 158 (2010), 136146.

D. S. Stones and I. M. Wanless,
Divisors of the number of Latin rectangles,
J. Combin. Th. Ser. A 117 (2010), 204215.

D. Bryant, J. Egan, B. Maenhaut and I. M. Wanless,
Indivisible plexes in latin squares,
Des. Codes Cryptogr. 52 (2009), 93105.

N.J. Cavenagh and I.M. Wanless,
Latin trades in groups defined on planar triangulations,
J. Algebraic Combin., 30 (2009), 323347.

J. A. Egan and I. M. Wanless,
Latin squares with no small odd plexes,
J. Combin. Designs, 16 (2008) 477492.

N. J. Cavenagh, C. Greenhill and I. M. Wanless,
The cycle structure of two rows in a random latin square,
Random Structures Algorithms 33 (2008) 286309.

B. D. McKay and I. M. Wanless,
A census of small latin hypercubes,
SIAM J. Discrete Math. 22, (2008) 719736.
(Click here
for data)

I. M. Wanless,
A computer enumeration of small latin trades,
Australas. J. Combin. 39, (2007) 247258. (Click
here
for data)

I. M. Wanless,
Transversals in latin squares,
Quasigroups Related Systems 15, (2007) 169190.

B. M. Maenhaut, I. M. Wanless and B. S. Webb,
Subsquarefree Latin
Squares of odd order,
Euro. J. Combin. 28, (2007) 322336.

B. D. McKay, J. C. McLeod and I. M. Wanless,
The number of transversals in a Latin square,
Des. Codes Cryptogr., 40, (2006) 269284. (Click
here
for data)

I. M. Wanless and B. S. Webb,
The Existence of Latin Squares without Orthogonal Mates,
Des. Codes Cryptogr.,
40, (2006) 131135.

D. Bryant, B. M. Maenhaut and I. M. Wanless,
New families of atomic Latin
squares and perfect onefactorisations,
J. Combin. Theory A 113, (2006) 608624.

B. D. McKay and I. M. Wanless,
On the number of Latin squares,
Ann. Combin. 9 (2005) 335344.

I. M. Wanless, Atomic Latin
squares based on cyclotomic orthomorphisms,
Electron. J. Combin. 12 (2005) R22. (Click
here
for extra data)

I. M. Wanless and E. C. Ihrig, Symmetries
that Latin squares inherit from 1factorizations,
J. Combin. Des., 13 (2005) 157172.

I. M. Wanless, Cycle switches
in Latin squares, Graphs Combin.
20 (2004), 545570.

I. M. Wanless,
A partial latin squares
problem posed by Blackburn, Bull. Inst. Comb. Appl.,
42 (2004), 7680.

I. M. Wanless,
Diagonally cyclic latin squares,
Euro. J. Combin., 25 (2004), 393413.

B. M. Maenhaut and I. M. Wanless,
Atomic Latin squares of order
eleven, J. Combin. Designs, 12 (2004), 1234.

M. VaughanLee and I. M. Wanless,
Latin squares and the HallPaige
conjecture, Bull. London Math. Soc.,
35 (2003) 15.

D. Bryant, B. M. Maenhaut and I. M. Wanless,
A family of perfect
factorisations of complete bipartite graphs,
J. Combin. Theory A 98 (2002) 328342.

I. M. Wanless,
A generalisation of transversals for Latin squares,
Electron.
J. Combin. 9 (2002) R12.

I. M. Wanless, Answers to questions by Denes on Latin
power sets, Euro. J. Combin., 22
(2001) 10091020.

I. M. Wanless, On McLeish's construction
for Latin squares without intercalates, Ars Combin.,
58 (2001) 313317.

I. M. Wanless, Latin squares with one subsquare,
J. Combin. Des. 9 (2001) 128146.

B. D. McKay and I. M. Wanless, Most Latin squares have many subsquares,
J. Combin. Theory Ser. A, 86 (1999) 323347.

I. M. Wanless, Permanents,
matchings and Latin rectangles (thesis abstract),
Bull. Austral. Math. Soc., 59 (1999)
169170.

I. M. Wanless, Perfect factorisations of bipartite graphs and
Latin squares without proper subrectangles,
Electron. J. Combin.
6 (1999) R9.
I've also worked on algebraic problems on groups, loops and
quasigroups (important algebraic structures whose Cayley tables are
latin squares). My papers in this area are:

M. Kinyon and I. M. Wanless,
Loops with exponent three in all isotopes,
submitted.

P. Vojtechovský and I. M. Wanless,
Closest multiplication tables of groups,
J. Algebra, 353 (2012), 261285.

A. Brouwer and I. M. Wanless,
Universally noncommutative loops,
Bull. Inst. Comb. Appl., 61 (2011), 113115.

D. Bryant, M. Buchanan and I. M. Wanless,
The spectrum for quasigroups
with cyclic automorphisms and additional symmetries,
Discrete Math. 309 (2009) 821833.
You can think of Latin squares as two dimensional permutations. I have
also worked on the more standard one dimensional permutations, giving rise
to:

C. J. Shallue and I. M. Wanless,
Permutation Polynomials and Orthomorphism Polynomials of Degree Six,
Finite Fields Appl. 20 (2013), 8492.

D. S. Stones and I. M. Wanless,
Compound orthomorphisms of the cyclic group,
Finite Fields Appl. 16 (2010), 277289.

P. J. Cameron and I. M. Wanless,
Covering radius
for sets of permutations,
Discrete Math. 293 (2005) 91109.
I have also done some work on Latin rectangles, which is what you get
if you chop a few rows off a Latin square. My principle interest here
has been in the following problem: For a rectangle of given
dimensions, what structure gives the most possible extensions to a
rectangle with one more row. Note that this is equivalent to a problem
of maximising the permanent and also to maximising perfect matchings
in regular bipartite graphs. My papers on this question are
listed in the next section.
The permanent of a matrix is like the determinant but without those
fiddly signs (+/) on each term. My papers on permanents are

D. Donovan, K. Johnson and I. M. Wanless,
Permanents and determinants of Latin squares,
J. Combin. Designs to appear (2015).

G.S. Cheon and I. M. Wanless,
Some results towards the Dittert conjecture on permanents,
Linear Algebra Appl., 436, (2012) 791801.

K. Pula, S.Z. Song and I. M. Wanless,
Minimum permanents on two faces of the polytope
of doubly stochastic matrices,
Linear Algebra Appl., 434 (2011) 232238.

I. M. Wanless,
On the BrualdiLiu conjecture for the even permanent,
Electron. J. Linear Algebra 17
(2008), 284286.

G.S. Cheon and I. M. Wanless,
An interpretation of the Dittert conjecture in terms of semimatchings,
Discrete Math., 307, (2007) 25012507.

I. M. Wanless,
Permanents, Chapter 31 in
Handbook of Linear Algebra (ed. L. Hogben),
Chapman & Hall/CRC (2007).

I. M. Wanless,
On Minc's sixth conjecture,
Linear and Multilinear Algebra,
55 (2007) 5763.

I. M. Wanless,
Addendum to Schrijver's work on minimum permanents,
Combinatorica, 26 (2006) 743745.

I. M. Wanless,
Permanents of matrices of signed ones,
Linear and Multilinear Algebra,
53 (2005) 427433.

G.S. Cheon and I. M. Wanless,
An update on Minc's survey of open problems involving permanents,
Lin. Alg. Appl., 403 (2005) 314342.

I. M. Wanless,
A lower bound on the maximum permanent in Λ_{n}^{k},
Linear Algebra Appl. 373 (2003), 153167.

I. M. Wanless, The HolensDokovic conjecture on permanents fails,
Linear Algebra Appl. 286 (1999) 273285.

I. M. Wanless, Maximising the permanent and complementary permanent of
(0,1) matrices with constant line sum,
Discrete Math., 205 (1999) 191205.

B. D. McKay and I. M. Wanless, Maximising the permanent of (0,1)
matrices and the number of extensions of Latin rectangles,
Electron. J. Combin.
5 (1998) R11.

I. M. Wanless, Permanents, matchings and Latin
rectangles (thesis abstract), Bull. Austral. Math. Soc.,
59 (1999) 169170.

I. M. Wanless, Jerrum's multistorey carpark, Austral. Math. Soc.
Gaz., 23 (1996) 193197.
A graph in this sense is a network of nodes and connections between
nodes. I have studied a few different problem areas, principally (i)
matchings and factorisations, (ii) asymptotic enumerations and random
graphs, (iii) graph games.
I've worked hard at matchings and onefactorisations in regular
bipartite graphs, which has resulted in a number of publications
already listed in previous sections (because these problems can be
expressed in terms of Permanents and Latin squares).
I have also sweated over asymptotic enumerations and random
graphs, with these results:

M. Minchenko and I. M. Wanless,
Spectral moments of regular graphs in terms of subgraph counts,
Lin. Alg. Appl. 446 (2014), 166176.

I. M. Wanless,
Counting matchings and treelike walks in regular graphs,
Combin. Probab. Comput. 19 (2010), 463480.

P. Lieby, B. D. McKay, J. C. McLeod and I. M. Wanless,
Subgraphs of random kedgecoloured kregular graphs,
Combin. Probab. Comput. 18 (2009), 533549.

B. D. McKay, I. M. Wanless and N. C. Wormald,
Asymptotic enumeration of graphs with a given upper bound on the
maximum degree, Combin. Probab. Comput. 11
(2002) 373392.

Z. C. Gao, I. M. Wanless and N. C. Wormald,
Counting 5connected planar triangulations, J. Graph Theory
38 (2001) 1835.

I. M. Wanless and N. C. Wormald,
Regular graphs with no homomorphisms onto cycles,
J. Combin. Theory B 82 (2001)
155160.

B. D. McKay, I. M. Wanless and N. C. Wormald,
The asymptotic number of graphs with a restriction on the maximum degree,
Electron. Notes Discrete Math. 5 (2000), 228230.
While at Melbourne Uni I also worked with
Stephen Taylor and Natashia Boland on an
amplifier placement problem, which resulted in...

S. Taylor, I. M. Wanless and N. L. Boland,
Distance domination and amplifier placement problems,
Australas. J. Combin. 34 (2006) 117136.
Finally, there's some `holiday' mathematics I've done on graph games.

I. M. Wanless, Path achievement games,
Australas. J. Combin. 23 (2001) 918.

I. M. Wanless, The periodicity of graph
games, Australas. J. Combin. 16 (1997)
113123.
Before closing this section I'll give you a question that I'd really
like solved. Suppose I have a kregular bipartite graph on 2n vertices
(where n>2k) and I know that among all such graphs my graph has the
most 4cycles. Does my graph necessarily contain a complete bipartite
component?
What else keeps me busy at work?
I am an editorinchief of the Electronic Journal of Combinatorics and
on the editorial board of the Journal of Combinatorial Designs
and a reviewer for
Zentralblatt.
From 20062009 and 20132014 I was the president of the Combinatorial
Mathematics Society of Australasia. In 2011, I was chair of the organising committee for 35ACCMCC.
Contact Information
Tel: +61 3 99054442
Email: Is of the standard form firstname.lastname@monash.edu
Snailmail:
School of Mathematical Sciences
Monash University
Vic 3800 Australia