Partition Identities for Ramanujan's Third-Order Mock Theta Functions

Abstract

We find two involutions on partitions that lead to partition identities for Ramanujan's third order mock theta functions φ(−q) and ψ(−q). We also give an involution for Fine's partition identity on the mock theta function f (q). The two classical identities of Ramanujan on third order mock theta functions are consequences of these partition identities. Our combinatorial constructions also apply to Andrews' generalizations of Ramanujan's identities.

On flushed partitions and concave compositions

“…We also write down a subscript n in the end to denote the size of the Durfee square. Take the partition λ = (11,11,11,9,7,5,5,4,4,3) for example, whose Young diagram is depicted in Figure 4, The Durfee symbol representation of λ is as follows: Suppose the Durfee square of a partition is of size n, we call a partition proper if its Durfee symbol has the same number of n's in both the top and bottom rows. All other partitions are improper partitions.…”

“…The number of improper partitions of n is denoted as IM P R(n). A typical example of a proper partition, λ = (11,11,11,9,7,5,5,4,4,3), has its Young diagram in Figure 4. In its Durfee symbol, both the top and bottom rows have two 5's.…”

“…The first proof of this identity is given by Watson in [17]. Another combinatorial proof was given by Chen, Ji and Liu in [11].…”