A Discontinuity in the Distribution of Fixed Point Sums

The quantity $f(n,r)$, defined as the number of permutations of the set $[n]=\{1,2,... n\}$ whose fixed points sum to $r$, shows a sharp discontinuity in the neighborhood of $r=n$. We explain this discontinuity and study the possible existence of other discontinuities in $f(n,r)$ for permutations. We generalize our results to other families of structures that exhibit the same kind of discontinuities, by studying $f(n,r)$ when ``fixed points'' is replaced by ``components of size 1'' in a suitable graph of the structure. Among the objects considered are permutations, all functions and set partitions.Comment: 1 figur

The Asymptotic Number of Irreducible Partitions

A partition of [1, n] = {1,..., n} is called irreducible if no proper subinterval of [1, n] is a union of blocks. We determine the asymptotic relationship between the numbers of irreducible partitions, partitions without singleton blocks, and all partitions when the block sizes must lie in some specified set

Three routes to the exact asymptotics for the one-dimensional quantum walk

We demonstrate an alternative method for calculating the asymptotic behaviour of the discrete one-coin quantum walk on the infinite line, via the Jacobi polynomials that arise in the path integral representation. This is significantly easier to use than the Darboux method. It also provides a single integral representation for the wavefunction that works over the full range of positions, $n,$ including throughout the transitional range where the behaviour changes from oscillatory to exponential. Previous analyses of this system have run into difficulties in the transitional range, because the approximations on which they were based break down here. The fact that there are two different kinds of approach to this problem (Path Integral vs. Schr\"{o}dinger wave mechanics) is ultimately a manifestation of the equivalence between the path-integral formulation of quantum mechanics and the original formulation developed in the 1920s. We discuss how and why our approach is related to the two methods that have already been used to analyse these systems.Comment: 25 pages, AMS preprint format, 4 figures as encapsulated postscript. Replaced because there were sign errors in equations (80) & (85) and Lemma 2 of the journal version (v3

Asymptotics of Some Convolutional Recurrences

Research supported by NSERC Research supported by NSERC and Canada Research Chair Program We study the asymptotic behavior of the terms in sequences satisfying recurrences of the form an = an−1 + ∑n−d k=d f(n,k)akan−k where, very roughly speaking, f(n,k) behaves like a product of reciprocals of binomial coefficients. Some examples of such sequences from map enumerations, Airy constants, and Painlevé I equations are discussed in detail. 1 Main results There are many examples in the literature of sequences defined recursively using a convolution. It often seems difficult to determine the asymptotic behavior of such sequences. In this note we study the asymptotics of a general class of such sequences. We prove subexponential growth by using an iterative method that may be useful for other recurrences. By subexponential growth we mean that, for every constant D> 1, an = o(D n