683 research outputs found
Statistical Distributions and q-Analogues of k-Fibonacci Numbers
Abstract We study q-analogues of k-Fibonacci numbers that arise from weighted tilings of an n × 1 board with tiles of length at most k. The weights on our tilings arise naturally out of distributions of permutations statistics and set partitions statistics. We use these q-analogues to produce q-analogues of identities involving k-Fibonacci numbers. This is a natural extension of results of the first author and Sagan on set partitions and the first author and Mathisen on permutations. In this paper we give general q-analogues of k-Fibonacci identities for arbitrary weights that depend only on lengths and locations of tiles. We then determine weights for specific permutation or set partition statistics and use these specific weights and the general identities to produce specific identities
Inversion Polynomials for Permutations Avoiding Consecutive Patterns
In 2012, Sagan and Savage introduced the notion of -Wilf equivalence for
a statistic and for sets of permutations that avoid particular permutation
patterns which can be extended to generalized permutation patterns. In this
paper we consider -Wilf equivalence on sets of two or more consecutive
permutation patterns. We say that two sets of generalized permutation patterns
and are -Wilf equivalent if the generating function for the
inversion statistic on the permutations that simultaneously avoid all elements
of is equal to the generating function for the inversion statistic on the
permutations that simultaneously avoid all elements of .
In 2013, Cameron and Killpatrick gave the inversion generating function for
Fibonacci tableaux which are in one-to-one correspondence with the set of
permutations that simultaneously avoid the consecutive patterns and
In this paper, we use the language of Fibonacci tableaux to study the
inversion generating functions for permutations that avoid where is
a set of five or fewer consecutive permutation patterns. In addition, we
introduce the more general notion of a strip tableaux which are a useful
combinatorial object for studying consecutive pattern avoidance. We go on to
give the inversion generating functions for all but one of the cases where
is a subset of three consecutive permutation patterns and we give several
results for a subset of two consecutive permutation patterns
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
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