Bose and Fermi walk configurations on planar graphs

Abstract

The number, fCn(H), of n-walk configurations of type C is investigated on certain two-rooted directed planar graphs H which will be always realized as plane graphs in R2. C may be Bose or Fermi as defined by Inui and Katori. Both types of configuration are collections of non-crossing walks which follow the directed paths between the roots of the plane graph H. In the case of configurations of Fermi type each walk may be included only once. The number fBosen(H) is shown to be a polynomial in n of degree nmax − 1 where nmax is the maximum number of walks in a Fermi configuration. The coefficient of the highest power of n in this polynomial is simply related to the number of maximal Fermi walk configurations. It is also shown that nmax = c(H) + 1 where c(H) is the number of finite faces on H. Extension of these results to multi-rooted graphs is also discussed. When H is the union of paths between two sites of the directed square lattice subject to various boundary conditions Kreweras showed that the number of Bose configurations is equal to the number of n-element multi-chains on segments of Young’s lattice. He expressed this number as a determinant the elements of which are polynomials in n. We evaluate this determinant by the method of LU decomposition in the case of ‘watermelon’ configurations above a wall. In this case the polynomial is a product of linear factors but on introducing a second wall the polynomial does not completely factorize but has a factor which is the number of watermelon configurations on the largest rectangular subgraph. The number of two-rooted ‘star’ configurations is found to be the product of the numbers of watermelon configurations on the three rectangular subgraphs into which it may be partitioned.