Alastair King’s scientific contributions

In the theory of random walks, it is notable that the central binomial coefficients 2nn count the number of walks of three different special types, which may be described as ‘balanced’, ‘nonnegative’ and ‘nonzero’. One of these coincidences is equivalent to the well-known convolution identity ∑ p+q=n 2p p2q q=2 2n · This article brings together several proofs of this ‘ubiquity of central binomial coefficients’ by presenting various relations between these classes of walks and combinatorial constructions that lead to the convolution identity. In particular, new natural bijections for the convolution identity based on the unifying idea of Catalan factorization are described.