A refined view of a curious identity for partitions into odd parts with designated summands

2025/05/27 by Fu, Shishuo, Sellers, James · 2 citations
#05A15 #05A17 #11P84 #33B10 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.21111

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 PDO(n).

Citations

• MacMahon's sums-of-divisors and their connection to multiple Eisenstein series
• An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands
• On the Number of Partitions with Designated Summands
• MacMahon's sum-of-divisors functions, Chebyshev polynomials, and\n Quasi-modular forms

Cited by

• Convolutive sequences, I: Through the lens of integer partition functions
• Convolutive sequences, II: Parametrizations

Related

• A rank of partitions with overline designated summands
• Marking and shifting a part in partition theorems
• A note on Andrews' partitions with parts separated by parity
• On the Number of Partitions with Designated Summands
• On partitions with even parts below odd parts