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# Category Archives: exposition

## Polynomials taking integer values

This is not a hot new result, just part of general mathematical culture. If a polynomial f(x) has integer coefficients, then its values at integer arguments are clearly integers. The converse is false; the simplest example is the polynomial x(x−1)/2. … Continue reading

Posted in exposition
Tagged difference polynomials, integers, polynomials, reciprocal polynomials
4 Comments

## Reciprocity

The cycle index of a finite permutation group is a multivariate polynomial (with one variable si for each index i not exceeding the degree of the group) which is the generating function for the numbers of cycles of different lengths … Continue reading

Posted in exposition
Tagged Bill Jackson, cycle polynomial, Jason Rudd, Jason Semeraro, orbital chromatic polynomial
1 Comment

## Two excursions

Last Friday, my friend Jan Kratochvil spoke to the Edinburgh Mathematical Society at the University of Stirling. I decided to go and hear him. The situation he talked about was typefied by this example. Suppose that you have a large … Continue reading

Posted in exposition, geography
Tagged Eildon Hills, geometric graphs, Jan Kratochvil, Ochil Hills, Stirling, Tweedbank
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## Road closures and idempotent-generated semigroups

The University of St Andrews is installing a biomass boiler, to provide hot water to heat University buildings, on the old paper mill site at Guardbridge. The water has to be piped four miles to St Andrews, and this big … Continue reading

## Rainbows in the plane

As usual in Lisbon, I have been working with João Araújo on semigroups. But sometimes research has unusual spin-offs, such as the following curious fact: Fact If the points of a projective plane are coloured with four colours (all of … Continue reading

## Imprimitive permutations in primitive groups

Asymptotics of permutation counts is a subject with a long history. The result that the proportion of permutations which are derangements (that is, have no fixed points) is very close to 1/e is one of the oldest results in enumerative … Continue reading

Posted in exposition
Tagged imprimitive, permutation, permutation group, primitive, Wielandt's conjecture
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## Curves

Just back from an enjoyable evening at Gresham College, where Jan van Maanen from Utrecht gave the BSHM–Gresham lecture marking the tercentenary of the death of Leibniz next month, preceded by warm-up acts by Snezana Lawrence and my colleague Kenneth … Continue reading