arXiv subjects

Peter J. Cameron

Publications and source records attributed to Peter J. Cameron.

114 records · Page 7Linked to original sources

arXiv2011

Combinatorial representations

This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families representable over a given alphabet. Then, we associate a rank function and a rank operator to any representation which help us determine some criteria for the functions used in a representation. While linearly representable matroids can be viewed as having representations via matrices with only one row, we conclude this paper by an investigation of representations via matrices with only two rows.

math.CO↗

arXiv2011

Dixon's Theorem and random synchronization

A transformation monoid on a set Omega is called synchronizing if it contains an element of rank 1 (that is, mapping the whole of Omega to a single point). In this paper, I tackle the question: given n and k, what is the probability that the submonoid of the full transformation monoid T_n generated by k random transformations is synchronizing? This question is analogous to Dixon's Theorem that two random permutations generate the symmetric or alternating group with high probability. Following the technique of Dixon's theorem, we need to analyse the maximal non-synchronizing submonoids of T_n. I develop a very close connection between transformation monoids and graphs, from which we obtain a description of non-synchronizing monoids as endomorphism monoids of graphs satisfying some very strong conditions. However, counting such graphs, and dealing with the intersections of their endomorphism monoids, seems difficult.

math.RA↗

arXiv2011

Counting false entries in truth tables of bracketed formulae connected by implication

In this paper we count the number of rows f_n with the value "false" in the truth tables of all bracketed formulae with n distinct variables connected by the binary connective of implication. We find a recurrence and an asymptotic formulae for f_n. We also show that the ratio of f_n to the total number of rows converges to \frac{3-\sqrt{3}}{6}.

math.CO↗

arXiv2011

A note on higher-dimensional magic matrices

We provide exact and asymptotic formulae for the number of unrestricted, respectively indecomposable, $d$-dimensional matrices where the sum of all matrix entries with one coordinate fixed equals 2.

math.CO↗

arXiv2008

Counting Defective Parking Functions

Suppose that $n$ drivers each choose a preferred parking space in a linear car park with $m$ spaces. Each driver goes to the chosen space and parks there if it is free, and otherwise takes the first available space with larger number (if any). If all drivers park successfully, the sequence of choices is called a parking function. In general, if $k$ drivers fail to park, we have a \emph{defective parking function} of \emph{defect} $k$. Let $\cp(n,m,k)$ be the number of such functions. In this paper, we establish a recurrence relation for the numbers $\cp(n,m,k)$, and express this as an equation for a three-variable generating function. We solve this equation using the kernel method, and extract the coefficients explicitly: it turns out that the cumulative totals are partial sums in Abel's binomial identity. Finally, we compute the asymptotics of $\cp(n,m,k)$. In particular, for the case $m=n$, if choices are made independently at random, the limiting distribution of the defect (the number of drivers who fail to park), scaled by the square root of $n$, is the Rayleigh distribution. On the other hand, in case $m=ω(n)$, the probability that all spaces are occupied tends asymptotically to one.

math.CO↗

arXiv2006

On the quantum chromatic number of a graph

We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the graph. After discussing this notion from first principles, we go on to establish relations with the clique number and orthogonal representations of the graph. We also prove several general facts about this graph parameter and find large separations between the clique number and the quantum chromatic number by looking at random graphs. Finally, we show that there can be no separation between classical and quantum chromatic number if the latter is 2, nor if it is 3 in a restricted quantum model; on the other hand, we exhibit a graph on 18 vertices and 44 edges with chromatic number 5 and quantum chromatic number 4.

quant-ph↗