Self-complementary ideals
The blue picture above can be thought of as a collection of cubes of unit side length stacked in a 6×6×6 box. If two cubes touch face to face, then the cube that is closest to the bottom back corner is defined to be the smaller one, which endows the set of 63 positions in the box with the structure of a partially ordered set (or poset for short). The set of occupied positions in the box forms an ideal, which means that if a position in the box is occupied, then we require all smaller positions to be occupied too. Informally, this means that the cubes are required to be stacked flush into the bottom back corner.
The ideal shown in blue has the additional property of being self-complementary. This means that if we reflect the picture in each of the three coordinate planes in turn, the net effect is to exchange the set of occupied positions with the set of unoccupied positions. Another way to think of this is that if we reflect the blue ideal in a mirror to form the red ideal shown above, then the red shape rotates to fill up the negative space in the blue shape perfectly. (This works because we are in three dimensions, and 3 is odd; in an even number of dimensions, we would skip the step of reflecting in a mirror.)
Self-complementary ideals can have additional symmetry properties. For example, the picture above and to the left shows a cyclically symmetric self-complementary ideal, which has the additional property of having threefold rotational symmetry. The picture above and to the right shows a totally symmetric self-complementary ideal, which has reflectional symmetry as well as threefold rotational symmetry.
These pictures come from the recent paper Flip Graphs on Self-Complementary Ideals of Chain Products by Serena An and Holden Mui. In this context, a chain consists of a set of numbers {1, 2, …, n}, ordered by ≤ in the usual way; such a chain is denoted by [n] for short. A product of chains would then be something like [a]×[b]×[c], where the numbers a, b, and c may or may not be the same. This can be thought of as an a×b×c box, whose positions are ordered relative to the bottom back corner as described in the first paragraph.
The authors introduce the notion of a flip on the self-complementary ideals of a chain product poset. The picture above shows an example of a flip corresponding to the poset [2]×[3]×[4]. The difference between the two ideals is that a single cube has been moved, but there are some restrictions. More specifically, (1) the cube being removed cannot be behind or below another cube, and (2) there is only one possible destination for the cube being moved so that the resulting ideal is self-complementary, namely the corresponding position in the negative space in the box. The flip graph is a picture of all possible self-complementary ideals of a given chain product poset, in which two ideals are joined by an edge if they differ by a flip. The picture below shows the flip graph on the chain product poset [2]×[3]×[4].
The authors prove a number of results about the flip graph. For example, they show that the graph is connected, which means it is always possible to move from one ideal in the graph to any other using a sequence of flips. They also prove a formula for the number of vertices in the graph in 1, 2, or 3 dimensions. Notice that a self-complementary ideal in a poset [a]×[b]×[c] cannot exist if all the numbers a, b, and c are odd. This is because a self-complementary ideal necessarily occupies precisely half the possible positions, which is impossible if a×b×c is odd.
The authors also study the analogues of flip graphs for cyclically symmetric self-complementary ideals (shown above) and totally symmetric self-complementary ideals (shown below). In these cases, one needs to move more than one cube at a time to retain the symmetry properties, so two ideals are joined by an edge if they differ by the smallest possible number of cubes. A thicker edge denotes a situation where relatively many cubes are moved at once.
Picture credits and relevant links
All the pictures are edited versions of pictures from the paper by Serena An and Holden Mui.
Wikipedia’s page on plane partitions has more information on some of the topics mentioned here.