A combinatorial proof for the positivity of the normalized Jacobi triple product tails

For $k\ge 1$, we prove that $$[q^n{z}^s]J_k(z,q)\ge 0,(n\ge 0,s\in \mathbb{Z})$$ for the normalized Jacobi triple product tails $$J_k(z,q)=\frac{\sum _{j=k}^{\mathrm{\infty }}(-1{)}^{j-k}{q}^{\left(\genfrac{}{}{0px}{}{j+1}{2}\right)}(z^{-j}+\cdots +z^j)}{(zq,q\mathrm{/}z;q{)}_{\mathrm{\infty }}}\mathrm{.}$$ This result not only implies Merca’s stronger nonnegativity conjecture on truncated Jacobi triple product series in full generality, but also yields infinite families of linear inequalities for two-colored partitions and partitions with parts in the residue classes $\pm S(\mathrm{m}\mathrm{o}\mathrm{d}R)$. We present a combinatorial proof wherein a sign-reversing involution reduces the normalized Jacobi triple product tails to the invariant subsets according to the generalized minimal-excludant of partitions. Furthermore, by combing an invertible lift operator on Frobenius arms with Konan’s size- and length-preserving bijection, an injection is constructed between the consecutive invariant subsets, which implies the coefficientwise positivity of the normalized Jacobi triple product tails.