384 research outputs found
Enumeration of saturated chains in Dyck lattices
We determine a general formula to compute the number of saturated chains in
Dyck lattices, and we apply it to find the number of saturated chains of length
2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of
Dyck lattices, which is the ratio between the total number of saturated chains
(of length 2 and 3) and the cardinality of the underlying poset.Comment: 9 page
Ascent Sequences Avoiding Pairs of Patterns
Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and SteingrÃmsson. In this paper, we consider ascent sequences of length n role= presentation style= display: inline; font-size: 11.2px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: Verdana, Arial, Helvetica, sans-serif; position: relative; \u3enn avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the ErdÅ‘s-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound
On the sub-permutations of pattern avoiding permutations
There is a deep connection between permutations and trees. Certain
sub-structures of permutations, called sub-permutations, bijectively map to
sub-trees of binary increasing trees. This opens a powerful tool set to study
enumerative and probabilistic properties of sub-permutations and to investigate
the relationships between 'local' and 'global' features using the concept of
pattern avoidance. First, given a pattern {\mu}, we study how the avoidance of
{\mu} in a permutation {\pi} affects the presence of other patterns in the
sub-permutations of {\pi}. More precisely, considering patterns of length 3, we
solve instances of the following problem: given a class of permutations K and a
pattern {\mu}, we ask for the number of permutations whose
sub-permutations in K satisfy certain additional constraints on their size.
Second, we study the probability for a generic pattern to be contained in a
random permutation {\pi} of size n without being present in the
sub-permutations of {\pi} generated by the entry . These
theoretical results can be useful to define efficient randomized pattern-search
procedures based on classical algorithms of pattern-recognition, while the
general problem of pattern-search is NP-complete
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