Optimal self-assembly of finite shapes at temperature 1 in 3D

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape $X \subset \mathbb{Z}^2$, there is a tile set that uniquely self-assembles into a 3D representation of a scaled-up version of $X$ at temperature 1 in 3D with optimal program-size complexity (the "program-size complexity", also known as "tile complexity", of a shape is the minimum number of tile types required to uniquely self-assemble it). Moreover, our construction is "just barely" 3D in the sense that it only places tiles in the $z = 0$ and $z = 1$ planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SICOMP 2007)

Congruences for 7 and 49-regular partitions modulo powers of 7

Let bk(n) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for b7(n) and b49(n). For example, for all j≥1 and n≥0, we prove that b7(72j−1n+3⋅72j−1−14)≡0(mod7j) and b49(7jn+7j−2)≡0(mod7j)