Partition Identities

A partition of a positive integer n (or a partition of weight n) is a non-decreasing sequence λ = (λ1, λ2, . . . , λk) of non-negative integers λi such that ∑k i=1 λi = n. The λi’s are the parts of the partition λ. Integer partitions are of particular interest in combinatorics, partly because many profound questions concerning integer partitions, solved and unsolved, are easily stated, but not easily proved. Even the most basic question “How many partitions are there of weight n?” has no simple solution. Remarkably, however, there are a variety of partition identities of the form “The number of partitions of n satisfying condition A is equal to the number of partitions of n satisfying condition B,” even though no simple formulas are known for the number of partitions of n satisfying A or B. The motivating example of such a partition identity is due to Euler: The number of partitions of n into distinct parts is equal to the number of partitions of n into odd parts. We will start here and our ultimate goal will be to examine a very recent development in the theory of partition identities: the so-called “Lecture Hall Partitions” of Bousquet-Mélou and Eriksson. Along the way, we will briefly visit some earlier results such as the Rogers-Ramanujan identities which have also extended the study of partition identities.