699 research outputs found
Restricted ascent sequences and Catalan numbers
Ascent sequences are those consisting of non-negative integers in which the
size of each letter is restricted by the number of ascents preceding it and
have been shown to be equinumerous with the (2+2)-free posets of the same size.
Furthermore, connections to a variety of other combinatorial structures,
including set partitions, permutations, and certain integer matrices, have been
made. In this paper, we identify all members of the (4,4)-Wilf equivalence
class for ascent sequences corresponding to the Catalan number
C_n=\frac{1}{n+1}\binom{2n}{n}. This extends recent work concerning avoidance
of a single pattern and provides apparently new combinatorial interpretations
for C_n. In several cases, the subset of the class consisting of those members
having exactly m ascents is given by the Narayana number
N_{n,m+1}=\frac{1}{n}\binom{n}{m+1}\binom{n}{m}.Comment: 12 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 Steingrímsson. 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 Erdős-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
Pattern Avoidance in Weak Ascent Sequences
In this paper, we study the pattern avoidance in weak ascent sequences,
giving some results for patterns of length 3. This is an analogous study to one
given by Duncan and Steingr\'imsson for ascent sequences. More precisely, we
provide systematically the generating functions for the number of weak ascent
sequences avoiding the patterns , and . We also
establish bijective connections between pattern-avoiding weak ascent sequences
and other combinatorial objects, such as compositions, upper triangular
01-matrices, and ordered trees. We leave as an open question the enumeration of
the remaining cases of length 3
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