Partitions with designated summands: k-tuples and their congruences
Abstract
Let Ak(n) and Bk(n) be the numbers of ordered k-tuples of partitions with designated summands, respectively with designated summands and all parts odd, of total size n. Using an exact description of the generating function of designated partitions after a $3$-dissection, we classify for every e the values of k for which Ak(3n+2) is divisible by 3e: this happens exactly when k≡0 or −5pmod3e if e≤4, and when 3e| k if e≥5. We obtain the corresponding exact $3$-adic and several $2$-adic results for Bk and for Ak along the progressions 4n+3, a recursion for the classes 3b(3n+s), and for k=2 the congruences A2(12n+10)≡ 27f14f63/f2pmod81 in generating-function form. We relate the numbers Ak(n) modulo $3$ and $9$ to overpartitions, sums of squares and triangular numbers, give group-action proofs of some of the congruences, and close with several conjectures supported by computation.
Related
- On Certain Congruences for Partition k-Tuples with $5$-Cores
- Congruences for Overcolored Partition k-tuples Restricted by Parity of the Parts
- Congruences for Overcubic Partition k-Tuples
- A uniform proof approach for congruences modulo 3 for partitions with k-colored odd parts
- Congruences modulo powers of $7$ for partitions with colored odd parts
BibTeX
@misc{yaqubi2026partitions,
title = {{Partitions with designated summands: $k$-tuples and their congruences}},
author = {Daniel Yaqubi and Madjid Mirzavziri},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05623},
eprint = {2611.05623},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05623},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Partitions with designated summands: k-tuples and their congruences
AU - Daniel Yaqubi
AU - Madjid Mirzavziri
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05623
UR - https://arxiv.org/abs/2611.05623
AB - Let A_k(n) and B_k(n) be the numbers of ordered k-tuples of partitions with designated summands, respectively with designated summands and all parts odd, of total size n. Using an exact description of the generating function of designated partitions after a $3$-dissection, we classify for every e the values of k for which A_k(3n+2) is divisible by 3^e: this happens exactly when k≡0 or −5pmod3^e if e≤4, and when 3^e| k if e≥5. We obtain the corresponding exact $3$-adic and several $2$-adic results for B_k and for A_k along the progressions 4n+3, a recursion for the classes 3^b(3n+s), and for k=2 the congruences A_2(12n+10)≡ 27f_1^4f_6^3/f_2pmod81 in generating-function form. We relate the numbers A_k(n) modulo $3$ and $9$ to overpartitions, sums of squares and triangular numbers, give group-action proofs of some of the congruences, and close with several conjectures supported by computation.
ER -