Bruce Westbury (Warwick)
In this talk I will explain some combinatorics associated
with the On-Line Sequence of Integers A059710.
The start of this sequence is: 1 0 1 1 4 10 35 ...
The n-th term of this sequence can be defined as the number of paths in a set of paths of length n in the triangular tiling of the plane. It can also be defined as the number of planar trivalent graphs in a disc with n boundary points such that there are no internal faces with less than six sides. The aim of the talk is to describe inverse bijections between these two sequences of finite sets. The motivation and inspiration comes from representation theory. The imaginary Cayley numbers are a seven dimensional irreducible representation of the automorphism group. The n-th term of this sequence is also the dimension of the space of invariant tensors in the n-th tensor power of this representation. However the aim of the talk is to present this as a combinatorial result. If we take the two dimensional representation of sl(2) then the corresponding sequence is the sequence of Catalan numbers. It is well-known that these count paths on the half-line or count the number of ways of connecting 2n points on the boundary of the disc with n non-intersecting arcs. There are also well-known bijections between these two sequences of finite sets.
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