arXiv subjects

Peter Cameron

Publications and source records attributed to Peter Cameron.

22 records · Page 2Linked to original sources

arXiv2008

Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by n generators and n(n-1)/2 relations for n less or equal then 7. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.

math.RA↗

arXiv2007

Asymptotic enumeration of 2-covers and line graphs

In this paper we find asymptotic enumerations for the number of line graphs on $n$-labelled vertices and for different types of related combinatorial objects called 2-covers. We find that the number of 2-covers, $s_n$, and proper 2-covers, $t_n$, on $[n]$ both have asymptotic growth $$ s_n\sim t_n\sim B_{2n}2^{-n}\exp(-\frac12\log(2n/\log n))= B_{2n}2^{-n}\sqrt{\frac{\log n}{2n}}, $$ where $B_{2n}$ is the $2n$th Bell number, while the number of restricted 2-covers, $u_n$, restricted, proper 2-covers on $[n]$, $v_n$, and line graphs $l_n$, all have growth $$ u_n\sim v_n\sim l_n\sim B_{2n}2^{-n}n^{-1/2}\exp(-[\frac12\log(2n/\log n)]^2). $$ In our proofs we use probabilistic arguments for the unrestricted types of 2-covers and and generating function methods for the restricted types of 2-covers and line graphs.

math.CO↗

arXiv2006

Asymptotics for incidence matrix classes

We define {\em incidence matrices} to be zero-one matrices with no zero rows or columns. A classification of incidence matrices is considered for which conditions of symmetry by transposition, having no repeated rows/columns, or identification by permutation of rows/columns are imposed. We find asymptotics and relationships for the number of matrices with $n$ ones in these classes as $n\to\infty$.

math.CO↗

arXiv2005

Asymptotic enumeration of incidence matrices

We discuss the problem of counting {\em incidence matrices}, i.e. zero-one matrices with no zero rows or columns. Using different approaches we give three different proofs for the leading asymptotics for the number of matrices with $n$ ones as $n\to\infty$. We also give refined results for the asymptotic number of $i\times j$ incidence matrices with $n$ ones.

math.CO↗