Sampling, conditionalizing, counting, merging, searching regular vines

We present a sampling algorithm for a regular vine on $n$ variables which starts at an arbitrary variable. A sampling order whose nested conditional probabilities can be written as products of (conditional) copulas in the vine and univariate margins is said to be implied by the regular vine. We show that there are $2^{n-1}$ implied sampling orders for any regular vine on $n$ variables. We show that two regular vines on $n$ and $m$ distinct variables can be merged in $2^{n+m-2}$ ways. This greatly simplifies the proof of the number of regular vines on $n$ variables. A notion of sampling proximity based on numbers of shared implied sampling orders is introduced, and we use this notion to define a heuristic for searching vine space that avoids proximate vines.