280 research outputs found
Generalised Pattern Avoidance
Recently, Babson and Steingrimsson have introduced generalised permutation
patterns that allow the requirement that two adjacent letters in a pattern must
be adjacent in the permutation. We consider pattern avoidance for such
patterns, and give a complete solution for the number of permutations avoiding
any single pattern of length three with exactly one adjacent pair of letters.
We also give some results for the number of permutations avoiding two different
patterns. Relations are exhibited to several well studied combinatorial
structures, such as set partitions, Dyck paths, Motzkin paths, and involutions.
Furthermore, a new class of set partitions, called monotone partitions, is
defined and shown to be in one-to-one correspondence with non-overlapping
partitions
Classification of bijections between 321- and 132-avoiding permutations
It is well-known, and was first established by Knuth in 1969, that the number
of 321-avoiding permutations is equal to that of 132-avoiding permutations. In
the literature one can find many subsequent bijective proofs of this fact. It
turns out that some of the published bijections can easily be obtained from
others. In this paper we describe all bijections we were able to find in the
literature and show how they are related to each other via ``trivial''
bijections. We classify the bijections according to statistics preserved (from
a fixed, but large, set of statistics), obtaining substantial extensions of
known results. Thus, we give a comprehensive survey and a systematic analysis
of these bijections. We also give a recursive description of the algorithmic
bijection given by Richards in 1988 (combined with a bijection by Knuth from
1969). This bijection is equivalent to the celebrated bijection of Simion and
Schmidt (1985), as well as to the bijection given by Krattenthaler in 2001, and
it respects 11 statistics--the largest number of statistics any of the
bijections respects
Transport of patterns by Burge transpose
We take the first steps in developing a theory of transport of patterns from
Fishburn permutations to (modified) ascent sequences. Given a set of pattern
avoiding Fishburn permutations, we provide an explicit construction for the
basis of the corresponding set of modified ascent sequences. Our approach is in
fact more general and can transport patterns between permutations and
equivalence classes of so called Cayley permutations. This transport of
patterns relies on a simple operation we call the Burge transpose. It operates
on certain biwords called Burge words. Moreover, using mesh patterns on Cayley
permutations, we present an alternative view of the transport of patterns as a
Wilf-equivalence between subsets of Cayley permutations. We also highlight a
connection with primitive ascent sequences.Comment: 24 pages, 4 figure
Sorting and preimages of pattern classes
We introduce an algorithm to determine when a sorting operation, such as
stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a
new proof of the description of West-2-stack-sortable permutations, that is
permutations that are completely sorted when passed twice through a stack, in
terms of patterns. We also solve the long-standing problem of describing
West-3-stack-sortable permutations. This requires a new type of generalized
permutation pattern we call a decorated pattern.Comment: 13 pages, 5 figures, to appear at FPSAC 201
Decomposing labeled interval orders as pairs of permutations
We introduce ballot matrices, a signed combinatorial structure whose
definition naturally follows from the generating function for labeled interval
orders. A sign reversing involution on ballot matrices is defined. We show that
matrices fixed under this involution are in bijection with labeled interval
orders and that they decompose to a pair consisting of a permutation and an
inversion table. To fully classify such pairs, results pertaining to the
enumeration of permutations having a given set of ascent bottoms are given.
This allows for a new formula for the number of labeled interval orders
Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3
Vincular and covincular patterns are generalizations of classical patterns
allowing restrictions on the indices and values of the occurrences in a
permutation. In this paper we study the integer sequences arising as the
enumerations of permutations simultaneously avoiding a vincular and a
covincular pattern, both of length 3, with at most one restriction. We see
familiar sequences, such as the Catalan and Motzkin numbers, but also some
previously unknown sequences which have close links to other combinatorial
objects such as lattice paths and integer partitions. Where possible we include
a generating function for the enumeration. One of the cases considered settles
a conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. We
also give an alternative proof of the classic result that permutations avoiding
123 are counted by the Catalan numbers.Comment: 24 pages, 11 figures, 2 table
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