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

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Let $R$ be a commutative ring with nonzero identity, let $\mathcal{I}\left(\mathcal{R}\right)$ be the set of all ideals of $R$ and $\delta :\mathcal{I}\left(\mathcal{R}\right)\to \mathcal{I}\left(\mathcal{R}\right)$ an expansion of ideals of $R$ defined by $I\mapsto \delta \left(I\right)$. We introduce the concept of $(\delta ,2)$-primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(\delta ,2)$-primary ideal if whenever $a,b\in R$ and $ab\in I$, then ${a}^{2}\in I$ or ${b}^{2}\in \delta \left(I\right)$. Our purpose is to extend the concept of $2$-ideals to $(\delta ,2)$-primary ideals of commutative rings. Then we investigate the basic properties of $(\delta ,2)$-primary ideals and also discuss the relations among $(\delta ,2)$-primary, $\delta $-primary and...

An infinite word is $S$-automatic if, for all $n\ge 0$, its $(n+1)$st letter is the output of a deterministic automaton fed with the representation of $n$ in the considered numeration system $S$. In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for $d\ge 2$, we state that a multidimensional infinite word $x:{\mathbb{N}}^{d}\to \Sigma $ over a finite alphabet $\Sigma $ is $S$-automatic for some abstract numeration...