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# Category Archives: exposition

## 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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## Another formula

Anyone who knows about the Principle of Inclusion and Exclusion will suspect that it can be applied to the relationship between sampling with and without replacement. In order to find the formula for sampling without replacement, we must be able … Continue reading