3,631 research outputs found
The number of {1243, 2134}-avoiding permutations
We show that the counting sequence for permutations avoiding both of the
(classical) patterns 1243 and 2134 has the algebraic generating function
supplied by Vaclav Kotesovec for sequence A164651 in The On-Line Encyclopedia
of Integer Sequences.Comment: 7 pages, 1 figur
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia
We construct a sequence of finite automata that accept subclasses of the
class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley
limit for the class of 4231-avoiding permutations is bounded below by 9.35.
This bound shows that this class has the largest such limit among all classes
of permutations avoiding a single permutation of length 4 and refutes the
conjecture that the Wilf-Stanley limit of a class of permutations avoiding a
single permutation of length k cannot exceed (k-1)^2.Comment: Submitted to Advances in Applied Mathematic
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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