THE NUMBER OF CYCLIC SUBGROUPS OF FINITE ABELIAN GROUPS AND MENON’S IDENTITY

We give a new formula for the number of cyclic subgroups of a finite abelian group. This is based on Burnside’s lemma applied to the action of the power automorphism group. The resulting formula generalises Menon’s identity.

Group Actions on Partitions

We introduce group actions on the integer partitions and their variances. Using generating functions and Burnside's lemma, we study arithmetic properties of the counting functions arising from group actions. In particular, we find a modulo 4 congruence involving the number of ordinary partitions and the number of partitions into distinct parts.

4. A combinatorial zoo

‘A combinatorial zoo’ presents a menagerie of combinatorial topics, ranging from the box (or pigeonhole) principle, the inclusion–exclusion principle, the derangement problem, and the Tower of Hanoi problem that uses combinatorics to determine how soon the world will end to Fibonacci numbers, the marriage theorem, generators and enumerators, and counting chessboards, which involves symmetry. The method used to average the numbers of colourings that remain unchanged by each symmetry in this latter problem is often called ‘Burnside’s lemma’. This concept has since been developed into a much more powerful result, which has been used to count a wide range of objects with a degree of symmetry, such as graphs and chemical molecules.