Asymptotic enumeration of lonesum matrices

Weighted Lonesum Matrices and Their Generating Function

A lonesum matrix is a (0, 1)-matrix uniquely determined by its column and row sums, and the sum of its all entries is called the “weight” of it. The generating function of numbers of weighted lonesum matrices of each weight is given. A certain explicit formula for the number of weighted lonesum matrices is also proved.

Asymptotic enumeration of orientations

We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c ·n−αγn, for suitable constants c, α, γ, with α = 4 for 2-orientations ...

Asymptotic Enumeration Methods

12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related

Asymptotic Enumeration Methods

Asymptotic enumeration of Latin rectangles

A k X n Latin rectangle is a k X n matrix with entries from {1,2,.. . , n} such that no entry occurs more than once in any row or column. (Thus each row is a permutation of the integers 1,2,..., n.) Let L(k, n) be the number of k x n Latin rectangles. An outstanding problem is to determine the asymptotic value of L(k, n) as n —• oo, with k bounded by a suitable function of n. The first attack o...