#### Loyola Marymount University

** Brian Macdonald and Jake Soldera **

*
Counting the Maximal Chains of Bond Lattices*: To each graph, one can associate a poset called the bond lattice of the graph. This poset encodes important combinatorial information about the graph including information about coloring of the graph. Brian and Jake received the Summer Undergraduate Research Program award from LMU to look at enumerating maximal chains in the bond lattice of a graph. They found several formulas for the number maximal chains in certain chordal graphs. You can see a poster they made about the research.

**Alex Abrams **

* The Shi Arrangement and Pointed Partitions*: The Shi arrangement is a hyperplane arrangement with many nice combinatorial properties. The poset of pointed partitions is a generalization of the partition lattice that comes from operad theory. It is well known that the intersection poset of the Shi arrangement and the poset of pointed partitions have the same characteristic polynomial. This suggests a connection between the two posets. Alex received the Summer Opportunities for Advanced Research award from LMU to look at this connection. They showed that the poset of pointed partitions is a quotient of the intersection poset of the Shi arrangement. This quotient sheds some light on the connection between these two objects. You can see a poster Alex made about the research.

** Rachel Meilak **

* Math and Magic: A variation of the de Bruijn Card Trick*: Many card tricks are based purely on mathematics. For her senior thesis, Rachel learned about the mathematics behind several different card tricks and then invented a new trick based on a previous one. In fact, she invented several related card tricks. Instead of giving away how the tricks work, here is an example of what the audience would see in one of the tricks. The magician gives a deck of cards to an audience member and asks them to cut the deck as many times as they wish. Then the magician hands the cards out to the audience. She randomly chooses a person and asks what card they have. Next, she asks another randomly chosen person what color card they have. To finish, the magician then reveals the card of a third audience member.

** Anna Salam **

*Parking Function Labelings of Noncrossing Bond Posets*: The noncrossing partition lattice is a poset with many nice combinatorial properties. Farmer, Hallam, and Smyth introduced a generalization of the noncrossing partition lattice that associates to each graph a poset called the noncrossing bond poset. Stanley showed a nice connection between the noncrossing partition lattice and parking functions. In particular, he showed that there is an edge labeling of the noncrossing partition lattice where the labels along the maximal chains of length *n* are exactly the parking functions of length *n*. For her senior thesis, Anna looked at which graphs give rise to noncrossing bond posets where the inherited parking function is an ER-labeling. She gave a complete classification of which graphs have this property, namely those that are labeled with a perfect elimination order. You can see a poster Anna made about the research.

#### Wake Forest University

**Shuya (Liz) Mao ** *
Perfect Elimination Orders of Simplicial Complexes*: For graphs it is well-known that a graph having a
perfect elimination ordering is equivalent to the graph being chordal. Hallam, Martin, and Sagan introduced a notion of perfect elimination orderings for pure simplicial complexes which generalized the definition for graphs.
However, in higher dimensions the problem of classification becomes harder. To better understand this classification, Liz looked at ways to put simplicial complexes together which preserves having a perfect elimination order. Moreover, she investigated the relationship between counting a certain type of subcomplex and the labeling of the
complex.

** Yuechen (Seraphina) Qi ** *
Counting Forests with Certain Patterns and Their Relationship to Coloring*: The chromatic polynomial of a graph is a tool that keeps track of
colorings of the graph. For any graph, the coefficients of its chromatic polynomial count certain types of subgraphs. For some classes graphs, there are labelings of the graph such that these subgraphs have nice descriptions.
Seraphina considered the case when these types of labelings give subgraphs which can be described by avoiding certain types of patterns. She developed several constructions of graphs to better understand this problem. Additionally, she
wrote Sage code to test if a graph possesses such a labeling. By proving some new results about the relationship between these labelings and the chromatic polynomial, she was able to make this algorithm more efficient.

** Matt Farmer **

For his master thesis, Matt studied a a noncrossing version of the bond lattice of a graph. He looked at edge labelings of this new poset in an attempt to find a
combinatorial description of the Möbius function as well as the characteristic polynomial of the new poset. You can find his thesis here.