Stephen C. Locke

The Game of Pathos: 10968

Editorial comment. M. L. Glasser notes that the result can be derived from equation (5.17) on page 28 of V. Kowalenko, N. E. Frankel, M. L. Glasser, and T. Taucher, Generalized Euler-Jacobi Inversion Formula and Asymptotics Beyond all Orders, LMS Lecture Note Series, no. 214 (Cambridge University Press, 1995). In addition, he points out that this reference provides similar expressions where the square root in the exponent is replaced by any rational power.

Graphs without Increasing Paths: 10572

Solution by Stephen C. Locke, Florida Atlantic University, Boca Raton, FL. Let V be an ndimensional binary vector space, and let S be a subspace of V. We may take S to be the row space of an m x n binary matrix M. Furthermore, we may assume that M is row-reduced, so that the leading 1 in each row is the only 1 in its column. We call these entries pivots. Construct a graph G whose vertices are the nonzero columns of M (G is empty when S has dimension 0). For each pivot element mi, 1,the vertex representing column j is adjacent to the vertex representing column r if mi,r #A 0. Thus all edges consist of a pivot column and a higher-indexed non-pivot column. In particular, G has no path i, j, k with i O an x /n !