Sign Patterns in a Two Colored Partition Companion series

We study two closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series $$S_1(q)=\sum _{n\ge 0}{s}_1(n)q^n=\sum _{a\ge 0}{q}^a(-q^{a+1};q{)}_{\mathrm{\infty }}^2$$ and its odd companion, denoted by $T_o(q)$. First, for the eta-normalized companion $$C(q)=(q;q{)}_{\mathrm{\infty }}{T}_o(q)=\sum _{n\ge 0}c(n)q^n,$$ we prove a strong form of the Andrews–El Bachraoui sign conjecture that $\mathrm{lim\hspace{0.17em}sup}c(n)=+\mathrm{\infty }$ and $\mathrm{lim\hspace{0.17em}inf}c(n)=-\mathrm{\infty }$. 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.