Sign Patterns in a Two Colored Partition Companion series
Abstract
We study three closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series S1(q)=∑n≥0s1(n)qn=∑a≥0qa(−qa+1;q)∞2 and its odd companion, denoted by To(q). First, for the eta-normalized companion C(q)=(q;q)∞ To(q)=∑n≥0c(n)qn, we prove a strong form of the Andrews--El Bachraoui sign conjecture that limsup c(n)=+∞ and liminf c(n)=−∞. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews--El Bachraoui congruence for s1(n) modulo 4. Third, we introduce a two-variable refinement of S1(q) which records the difference of the lengths of the two colored components. Using Ramanujan's partial theta identity and heat-type equations, we obtain a closed formula for all of its even moments, and we show that the eta-normalized moments satisfy termwise congruences modulo p2k for every prime p≡5pmod6.
Related
- Efficient Computation and Congruences for Colored Partition Functions
- Efficient Solution of Third-Order ODEs with Diagonally Implicit RK Method Using Tri-Colored Tree and B-series Analysis
- Two Companion q-Series Identities and Their Combinatorics
- On sign patterns of Fourier coefficients of theta quotients
- Prescribed Turán sign patterns for inverse Kazhdan-Lusztig polynomials of matroids
BibTeX
@misc{dhar2026sign,
title = {{Sign Patterns in a Two Colored Partition Companion series}},
author = {Aritram Dhar and Ankush Goswami and Mohit Tripathi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.10576},
eprint = {2607.10576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.10576},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sign Patterns in a Two Colored Partition Companion series
AU - Aritram Dhar
AU - Ankush Goswami
AU - Mohit Tripathi
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2607.10576
UR - https://arxiv.org/abs/2607.10576
AB - We study three closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series S_1(q)=∑_n≥0s_1(n)q^n=∑_a≥0q^a(−q^a+1;q)_∞^2 and its odd companion, denoted by T_o(q). First, for the eta-normalized companion C(q)=(q;q)_∞ T_o(q)=∑_n≥0c(n)q^n, we prove a strong form of the Andrews--El Bachraoui sign conjecture that limsup c(n)=+∞ and liminf c(n)=−∞. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews--El Bachraoui congruence for s_1(n) modulo 4. Third, we introduce a two-variable refinement of S_1(q) which records the difference of the lengths of the two colored components. Using Ramanujan's partial theta identity and heat-type equations, we obtain a closed formula for all of its even moments, and we show that the eta-normalized moments satisfy termwise congruences modulo p^2k for every prime p≡5pmod6.
ER -