PreprintCC BY 4.0arXiv
A Combinatorial Proof of Partition Identities for the Number of Distinct Part Sizes
Abstract
In 1984, Bressoud and Subbarao established an interesting weighted partition identity connecting the partition function and the divisor function. Recently, the first author, jointly with Bhoria, Eyyunni, and Maji, extended the work of Bressoud and Subbarao and obtained the following analogous weighted partition identity. For any n ∈ ℕ, they obtained p(2)(n) = ∑π∈ 𝒟(n) (−1) # (π) s(π) ( ℓ(π) − s(π) ). In the present paper, we provide a combinatorial proof of the above identity and extend the result for p(k)(n), where p(k)(n) denotes the number of partitions of n into exactly k distinct part sizes.
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BibTeX
@misc{agarwal2026combinatorial,
title = {{A Combinatorial Proof of Partition Identities for the Number of Distinct Part Sizes}},
author = {Archit Agarwal and Meghali Garg},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18371},
eprint = {2610.18371},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18371},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Combinatorial Proof of Partition Identities for the Number of Distinct Part Sizes
AU - Archit Agarwal
AU - Meghali Garg
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18371
UR - https://arxiv.org/abs/2610.18371
AB - In 1984, Bressoud and Subbarao established an interesting weighted partition identity connecting the partition function and the divisor function. Recently, the first author, jointly with Bhoria, Eyyunni, and Maji, extended the work of Bressoud and Subbarao and obtained the following analogous weighted partition identity. For any n ∈ ℕ, they obtained p^(2)(n) = ∑_π∈ 𝒟(n) (−1)^ # (π) s(π) ( ℓ(π) − s(π) ). In the present paper, we provide a combinatorial proof of the above identity and extend the result for p^(k)(n), where p^(k)(n) denotes the number of partitions of n into exactly k distinct part sizes.
ER -