116 research outputs found
Counting occurrences of patterns in permutations
We develop a new, powerful method for counting elements in a {\em multiset.}
As a first application, we use this algorithm to study the number of
occurrences of patterns in a permutation. For patterns of length 3 there are
two Wilf classes, and the general behaviour of these is reasonably well-known.
We slightly extend some of the known results in that case, and exhaustively
study the case of patterns of length 4, about which there is little previous
knowledge. For such patterns, there are seven Wilf classes, and based on
extensive enumerations and careful series analysis, we have conjectured the
asymptotic behaviour for all classes.
Finally, we investigate a proposal of Blitvi\'c and Steingr\'imsson as to the
range of a parameter for which a particular generating function formed from the
occurrence sequences is itself a Stieltjes moment sequence
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