Counting canonical partitions in the random graph

Abstract

Joyce trees have concrete realizations as J-trees of sequences of 0’s and 1’s. Algorithms are given for computing the number of minimal height J-trees of d-ary sequences with n leaves and the number of them with minimal parent passing numbers to obtain polynomials ρ n (d) for the full collection and α n (d) for the subcollection. The number of traditional Joyce trees is the tangent number α n (1); α n (2) is the number of cells in the canonical partition by Laflamme, Sauer and Vuksanovic of n-element subsets of the infinite random (Rado) graph; and ρ n (2) is the number of weak embedding types of rooted n-leaf J-trees of sequences of 0’s and 1’s.