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# Tag Archives: partitions

## The existential transversal property

One of the first things that João Araújo introduced me to when we started collaborating, after synchronization, was the universal transversal property: a permutation group G on the set {1,…,n} has the k-universal transversal property (k-ut for short) if, given … Continue reading

Posted in exposition
Tagged partitions, permutation groups, regular semigroups, transversals
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## A puzzle for you

The Petersen graph is perhaps the most famous graph of all. It has ten vertices, fifteen edges, valency 3, and no triangles. Since the complete graph on ten vertices has 45 edges and valency 9, one might ask whether the … Continue reading

## Partition homogeneity

This post discusses a paper by Jorge André, João Araújo and me, which has just been accepted for the Journal of Algebra. There is a slight tangle to the history of this paper, which I want to describe. First a … Continue reading

Posted in exposition, history
Tagged homogeneity, partitions, semigroups, symmetric groups, transitivity
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## G. C. Steward lectures 2008

While I was uploading lecture notes, I also put on the page the notes from my G. C. Steward lectures at Gonville and Caius College in 2008. You can find them here. I spent the first half of 2008 in … Continue reading

Posted in history, Lecture notes, Neill Cameron artwork
Tagged automata, bagali polo, Euler, Latin squares, line graphs, magic squares, Moebius inversion, OEIS, On-line Encyclopedia of Integer Sequences, parking functions, partitions, root systems, statistics, Sudoku, synchronization, Tehran
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## A cliff

The “combinatorial explosion” is a well-known phenomenon. I recently came across a very dramatic example of it. I was trying to compute the function F(n,k), defined to be the maximum of |S|×|P|, over all sets S of k-subsets and all … Continue reading

Posted in exposition
Tagged combinatorial explosion, computers, GAP, GRAPE, Leonard Soicher, partitions, subsets, transversals
4 Comments

## Combinatorics Ancient and Modern

Towards the end of 2011, I posted a paper on the arXiv with the title “Aftermath”. A correspondent wondered if this was a Borgesian game, the final chapter to a nonexistent book. Happily, it was not so. Combinatorics Ancient and … Continue reading

Posted in books
Tagged block designs, graph theory, history of mathematics, Latin squares, partitions
1 Comment