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# Category Archives: exposition

## Representing the Fano matroid

In my lecture today I proved that the Fano matroid is representable over a field F if and only the characteristic of F is 2. There is a proof of this using only the classical theorems of Ceva and Menelaus … Continue reading

Posted in exposition
Tagged Ceva's theorem, Euclidean geometry, Fano plane, matroid, Menelaus' theorem, representation
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## 1247

Yesterday my colleague Collin Bleak sent me an email containing the number 1247. I will explain why this made me so pleased. Automata Our problem concerns automata. If you know Collin, you may suspect that it has something to do … Continue reading

## The combinatorial hierarchy

Following up a conversation with John Amson at Rufflets Hotel, just outside St Andrews, I was led to a paper on combinatorial physics: Ted Bastin, H. Pierre Noyes, John Amson, Clive W. Kilmister: On the physical interpretation and the mathematical … Continue reading

Posted in exposition
Tagged combinatorics, fine structure constant, John Amson, linear independence, physics
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## Orbital combinatorics

Yesterday I went to Edinburgh to give a colloquium talk about synchronization, including the recent stuff about butterflies. The day before, I had discussed Artur Schäfer’s work with him, and he expressed a hope that if he went to the … Continue reading

## From M12 to M24

On 20 January 2015, Paul Hjelmstad posted the following question on the GAP forum: Is there an easy way to generate a Steiner system S(5,8,24) for the Mathieu Group M24, if a Steiner system S(5,6,12) for the Mathieu Group M12 … Continue reading

Posted in exposition
Tagged Graham Higman, Jack van Lint, Mathieu group, outer automorphism, Paul Hjelmstad, Steiner system
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## Chains of semigroups

I have written here about the lovely formula for the length of the longest chain of subgroups in the symmetric group Sn: take n, increase it by 50% (rounding up if necessary), subtract the number of ones in the base … Continue reading

## Bijective proofs

A fourth proof Last month I described three proofs of the formula for the number of ways to choose k objects from a set of n, if repetition is allowed and order is not significant; it is the same as … Continue reading

Posted in exposition, open problems
Tagged bijections, Catalan numbers, Catalan objects, Dima Fon-Der-Flaass, permutations, sampling
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