On Pattern-avoiding chord diagrams

In a recent work of Lukas Nabergall and Karen Yeats (University of Waterloo) it is shown that a generic class of tree-like functional equations are solved in terms of weighted connected chord diagrams that forbid certain subdiagrams. Particularly, forbidding cycles is of interest. Accordingly, a number of conjectures and open questions were stated, regarding these special classes of chord diagrams. In this talk, some of these conjectures are proven using (elementary) enumerative methods. Specifically, We will prove that $C_n(T_{\geq 3},B_{\geq 3})$, the number of connected chord diagrams forbidding top and bottom cycles for induced subdiagrams, is the same as $Int(L^K_{n−1})$, the number of intervals in the Kreweras Lattice of size $n − 1$, which is $\frac{1}{2n+1}\binom{3n}{n}$. We will also give an argument on how to count the intriguing $C_n(B_{\geq 3})$. The proofs turned out to simplify existing parallel results.