8 research outputs found
Pattern avoidance in compositions and multiset permutations
We study pattern avoidance by combinatorial objects other than permutations,
namely by ordered partitions of an integer and by permutations of a multiset.
In the former case we determine the generating function explicitly, for integer
compositions of n that avoid a given pattern of length 3 and we show that the
answer is the same for all such patterns. We also show that the number of
multiset permutations that avoid a given three-letter pattern is the same for
all such patterns, thereby extending and refining earlier results of Albert,
Aldred et al., and by Atkinson, Walker and Linton. Further, the number of
permutations of a multiset S, with a_i copies of i for i = 1, ..., k, that
avoid a given permutation pattern in S_3 is a symmetric function of the a_i's,
and we will give here a bijective proof of this fact first for the pattern
(123), and then for all patterns in S_3 by using a recently discovered
bijection of Amy N. Myers.Comment: 8 pages, no figur
Pattern Avoidance in Poset Permutations
We extend the concept of pattern avoidance in permutations on a totally
ordered set to pattern avoidance in permutations on partially ordered sets. The
number of permutations on that avoid the pattern is denoted
. We extend a proof of Simion and Schmidt to show that for any poset , and we exactly classify the posets for which
equality holds.Comment: 13 pages, 1 figure; v2: corrected typos; v3: corrected typos and
improved formatting; v4: to appear in Order; v5: corrected typos; v6: updated
author email addresse
Sorting permutations with pattern-avoiding machines
In this work of thesis we introduce and study a new family of sorting
devices, which we call pattern-avoiding machines. They consist of two stacks in
series, equipped with a greedy procedure. On both stacks we impose a static
constraint in terms of pattern containment: reading the content from top to
bottom, the first stack is not allowed to contain occurrences of a given
pattern , whereas the second one is not allowed to contain occurrences
of . By analyzing the behavior of pattern-avoiding machines, we aim to gain
a better understanding of the problem of sorting permutations with two
consecutive stacks, which is currently one of the most challenging open
problems in combinatorics.Comment: PhD Thesis, 137 page
Priority Queues and Multisets
A priority queue, a container data structure equipped with the operations insert and delete-minimum, can re-order its input in various ways, depending both on the input and on the sequence of operations used. If a given input oe can produce a particular output ø then (oe; ø ) is said to be an allowable pair. It is shown that allowable pairs on a fixed multiset are in one-to-one correspondence with certain k-way trees and, consequently, the allowable pairs can be enumerated. Algorithms are presented for determining the number of allowable pairs with a fixed input component, or with a fixed output component. Finally, generating functions are used to study the maximum number of output components with a fixed input component, and a symmetry result is derived