Abstract:
We consider permutations pi of {1,...,n} as chord diagrams, where the elements label the vertices of a regular n-gon, and there is a directed arc from i to pi(i) for each element i. We can "rotate" a permutation by rotating its chord diagram.
As one of our main results we show that there must exist a statistic on permutations of {1,...,n} that has the same distribution as the length of the longest increasing subsequence, but is invariant under rotation.
The proof uses a little combinatorial representation and invariant theory, and some calculus. It appears non-trivial to exhibit the statistic explicitly.
The main motivation of a two-element subset of the authors is to find a "web" basis (in the sense of Kuperberg) for the adjoint representation of the general linear group.
This is joint work with Per Alexandersson, Stephan Pfannerer & Joakim Uhlin.