### $(-1)$-enumeration of self-complementary plane partitions.

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Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let ${p}_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let $p{*}_{k,m}(j,n)$ be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then $p{*}_{k,m}(j,n)={p}_{k,m}(j,n)$.

Let $k\ge 5$ be an odd integer and $\eta $ be any given real number. We prove that if ${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$, $\mu $ are nonzero real numbers, not all of the same sign, and ${\lambda}_{1}/{\lambda}_{2}$ is irrational, then for any real number $\sigma $ with $0<\sigma <1/\left(8\vartheta \right(k\left)\right)$, the inequality $$|{\lambda}_{1}{p}_{1}^{2}+{\lambda}_{2}{p}_{2}^{2}+{\lambda}_{3}{p}_{3}^{2}+{\lambda}_{4}{p}_{4}^{2}+\mu {p}_{5}^{k}+\eta |<{\left(\underset{1\le j\le 5}{max}{p}_{j}\right)}^{-\sigma}$$ has infinitely many solutions in prime variables ${p}_{1},{p}_{2},\cdots ,{p}_{5}$, where $\vartheta \left(k\right)=3\times {2}^{(k-5)/2}$ for $k=5,7,9$ and $\vartheta \left(k\right)=[({k}^{2}+2k+5)/8]$ for odd integer $k$ with $k\ge 11$. This improves a recent result in W. Ge, T. Wang (2018).