Graduate Thesis Or Dissertation
 

Asymptotic enumeration for combinatorial structures

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  • There are many combinatorial structures which can be regarded as complexes of certain basic blocks. Familiar examples are involutions, finite graphs, and Stirling numbers of the first and second kind. Generating functions for these complexes have special forms relating the number of basic blocks to the number of complexes. Previous results on asymptotic enumeration have yielded implicit formulae for the mean and variance of the counts of blocks, together with Gaussian approximations with these parameters. We will provide explicit formulae for these basic quantities, together with large deviation estimates for decay rates from the mean. As a result we obtain laws of large numbers and, under certain convexity conditions, a new approach to Gaussian approximations previously obtained by more cumbersome analysis. In regard to applications, we will provide a new connection between these combinatorial structures and a class of self-similar trees. Extensive data analysis of naturally occurring river networks has been shown by hydrologists to be nicely modeled by this class of self-similar trees. Through this connection, we will obtain new results on the number of paths of various Horton orders.
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