Abstract
Combinatorics We provide a rather general asymptotic scheme for combinatorial parameters that asymptotically follow a discrete double-exponential distribution. It is based on analysing generating functions Gh(z) whose dominant singularities converge to a certain value at an exponential rate. This behaviour is typically found by means of a bootstrapping approach. Our scheme is illustrated by a number of classical and new examples, such as the longest run in words or compositions, patterns in Dyck and Motzkin paths, or the maximum degree in planted plane trees.
Highlights
One encounters the situation that one obtains a sequence of generating functions Gh(z), depending on a parameter h whose distribution is to be studied, such that the dominant singularity ζh of Gh converges to a value ζ as h → ∞, and ζh − ζ decreases exponentially with h
Remark 4 The conditions are quite natural for combinatorial applications, but it is possible to modify them in various ways
Thereafter, we consider a variety of combinatorial examples to which this general asymptotic scheme can be applied
Summary
One encounters the situation that one obtains a sequence of generating functions Gh(z), depending on a parameter h whose distribution is to be studied, such that the dominant singularity ζh of Gh converges to a value ζ as h → ∞, and ζh − ζ decreases exponentially with h. The archetypical example is probably the distribution of the longest sequence of 1’s in a random 0-1-sequence: Let Gh(z) denote the generating function for 0-1-sequences with the property that there is no sequence of more than h consecutive 1’s. Behaviour in the dominant singularity typically leads to a discrete limit law analogous to the Gumbel distribution, and to periodic fluctuations. Remark 4 The conditions are quite natural for combinatorial applications, but it is possible to modify them in various ways (e.g., by allowing further terms in the asymptotic expansions with exponents between α and α + 1, with additional assumptions on the coefficients.) Proofs of these two theorems are provided . Thereafter, we consider a variety of combinatorial examples to which this general asymptotic scheme can be applied
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