A refined view of a curious identity for partitions into odd parts with designated summands
Shishuo Fu, James Sellers
TL;DR
The paper investigates partitions with designated summands, focusing on a refined counting framework that tracks the number of distinct part sizes in the odd-restricted PDO family. By leveraging a Chebyshev-polynomial–based generating function $G(x,q)$ and a two-dissection analysis, it establishes a symmetric GxGy identity under a specific change of variables and translates this into a two-parameter refinement via generating functions $P_1(x,y,q)$ and $P_2(x,y,q)$, with the key result $[q^{2n}]P_1(x,y,q)=[q^n]P_2(x,y,q)$. This leads to a refined combinatorial Interpretation of PDO partitions and a partition-theoretic equivalence between refined counts of $ ext{PDO}$ partitions and PDO-pairs, including explicit Euler-type product formulas for the refined generating functions and a special-case bijection when $y=0$. The work connects to MacMahon-type divisor sums, theta functions, and Chebyshev identities, providing both analytic and combinatorial insights into the structure of designated summands and their refinements.
Abstract
In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called partitions with designated summands. These are constructed by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In the same work, they also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresponding function by $\mathrm{PDO}(n)$.