In this short note, we use compositions to study the partition perimeter, a statistic defined to be one less than the sum of the number of parts and the largest part. This leads to some generalizations of known theorems. Our main result is a combinatorial proof that for m > 2 and n > m, there are strictly more m-distinct partitions than m-regular partitions with perimeter n, which provides an affirmative answer to a question from a recent paper of Amdeberhan et al. Additional refinements and applications of this are still being investigated.

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In his classic book, Topics in Analytic Number Theory, H. Rademacher posed a natural conjecture concerning the generating function for p(n), the number of partitions of n. In this paper we undertake a systematic study of an expansion technique that has its genesis in the work of Cayley. We apply this to the Rademacher conjecture, and obtain the first positive result providing theoretical evidence for the conjecture.

George Andrews recently proved a new identity between the cardinalities of the set of Semi-Fibonacci partitions and the set of partitions into powers of two with all parts appearing an odd number of times. This paper extends the identity to the set of Semi-m-Fibonacci partitions of n and the set of partitions of n into powers of m in which all parts appear with multiplicity not divisible by m. We also give a new characterization of Semi-m-Fibonacci partitions and some congruences satisfied by the associated number sequence.

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