Abstract

Counting related to representation theory and symmetric functions can be framed as generating functions given by Toeplitz determinants. Prime examples are counting all permutations with no long increasing subsequence or lattice paths in last passage percolation. Intricate scaling limits of those generating functions have been used, e.g., in the seminal work by Baik/Deift/Johansson, to obtain asymptotic formulae in terms of random matrix theory. In this talk, we address the question whether generating functions can be used to numerically extract the counts in a mesoscopic regime where combinatorial methods are already infeasible and the random matrix asymptotics is still too inaccurate. The stable computation of the counts by means of complex analysis is possible, indeed, and can be explained by the theory of Hayman admissibility. As a bonus track from complex analysis, the numerical evaluation of the Toeplitz determinant itself has to be stabilized by a variant of the Borodin-Okounkov formula based on the Wiener-Hopf factorization. This way, we obtain, e.g., exact 1135-digit counts in permutations of order 500 or, by taking Hayman’s famous generalization of Stirling’s formula at face value, a blazingly fast, surprisingly robust and accurate numerical asymptotics.