94,452 research outputs found
Place-difference-value patterns: A generalization of generalized permutation and word patterns
Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson
introduced the notion of a "generalized permutation pattern" (GP) which
generalizes the concept of "classical" permutation pattern introduced by Knuth
in 1969. The invention of GPs led to a large number of publications related to
properties of these patterns in permutations and words. Since the work of
Babson and Steingrimsson, several further generalizations of permutation
patterns have appeared in the literature, each bringing a new set of
permutation or word pattern problems and often new connections with other
combinatorial objects and disciplines. For example, Bousquet-Melou et al.
introduced a new type of permutation pattern that allowed them to relate
permutation patterns theory to the theory of partially ordered sets.
In this paper we introduce yet another, more general definition of a pattern,
called place-difference-value patterns (PDVP) that covers all of the most
common definitions of permutation and/or word patterns that have occurred in
the literature. PDVPs provide many new ways to develop the theory of patterns
in permutations and words. We shall give several examples of PDVPs in both
permutations and words that cannot be described in terms of any other pattern
conditions that have been introduced previously. Finally, we raise several
bijective questions linking our patterns to other combinatorial objects.Comment: 18 pages, 2 figures, 1 tabl
Generalized permutation patterns - a short survey
An occurrence of a classical pattern p in a permutation Ļ is a subsequence of Ļ whose letters are in the same relative order (of size) as those in p. In an occurrence of a generalized pattern, some letters of that subsequence may be required to be adjacent in the permutation. Subsets of permutations characterized by the avoidanceāor the prescribed number of occurrencesā of generalized patterns exhibit connections to an enormous variety of other combinatorial structures, some of them apparently deep. We give a short overview of the state of the art for generalized patterns
Introduction to Partially Ordered Patterns
We review selected known results on partially ordered patterns (POPs) that
include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified)
maxima and minima) in permutations, the Horse permutations and others. We
provide several (new) results on a class of POPs built on an arbitrary flat
poset, obtaining, as corollaries, the bivariate generating function for the
distribution of peaks (valleys) in permutations, links to Catalan, Narayna, and
Pell numbers, as well as generalizations of few results in the literature
including the descent distribution. Moreover, we discuss q-analogue for a
result on non-overlapping segmented POPs. Finally, we suggest several open
problems for further research.Comment: 23 pages; Discrete Applied Mathematics, to appea
Topics in algorithmic, enumerative and geometric combinatorics
This thesis presents five papers, studying enumerative and
extremal problems on combinatorial structures.
The first paper studies Forman's discrete Morse theory in the case where a group acts on the underlying complex. We generalize the notion of a Morse matching, and obtain a theory that can be used to simplify the description of the G-homotopy type of a simplicial complex. As an application, we determine the S_2xS_{n-2}-homotopy type of the complex of non-connected graphs on n nodes. In the introduction, connections are drawn between the first paper and the evasiveness conjecture for monotone graph properties.
In the second paper, we investigate Hansen polytopes of split graphs. By applying a partitioning
technique, the number of nonempty faces is counted, and in particular we confirm
Kalai's 3^d-conjecture for such polytopes. Furthermore, a characterization of
exactly which Hansen polytopes are also Hanner polytopes is given. We end by
constructing an interesting class of Hansen polytopes having very few faces and
yet not being Hanner.
The third paper studies the problem of packing a pattern as densely as possible into compositions. We are able to find the
packing density for some classes of generalized patterns, including all the three letter patterns.
In the fourth paper, we present combinatorial proofs of the enumeration of derangements with descents in prescribed positions. To this end, we
consider fixed point lambda-coloured permutations, which are easily
enumerated. Several formulae regarding these numbers are given, as
well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, the event that pi has descents in a set S of positions is positively correlated with the event that pi is a derangement, if pi is chosen uniformly in S_n.
The fifth paper solves a partially ordered generalization of the famous secretary problem. The elements of a finite nonempty partially ordered set are exposed in uniform random order to a selector who, at any given time, can see the relative order of the exposed elements. The selector's task is to choose online a maximal element. We describe a strategy for the general problem that achieves success probability at least 1/e for an arbitrary partial order, thus proving that the linearly ordered set is at least as difficult as any other instance of the problem. To this end, we define a probability measure on the maximal elements of an arbitrary partially ordered set, that may be interesting in its own right
Charge Ordered RVB States in the Doped Cuprates
We study charge ordered d-wave resonating valence bond states (dRVB) in the
doped cuprates, and estimate the energies of these states in a generalized model by using a renormalized mean field theory. The long range Coulomb
potential tends to modulate the charge density in favor of the charge ordered
RVB state. The possible relevance to the recently observed
checkerboard patterns in tunnelling conductance in high cuprates is
discussed.Comment: 4 pages, 4 figures, 3 table
Influence of long-range interactions on charge ordering phenomena on a square lattice
Usually complex charge ordering phenomena arise due to competing
interactions. We have studied how such ordered patterns emerge from the
frustration of a long-ranged interaction on a lattice. Using the lattice gas
model on a square lattice with fixed particle density, we have identified
several interesting phases; such as a generalization of Wigner crystals at low
particle densities and stripe phases at densities in between rho = 1/3 and rho
= 1/2. These stripes act as domain walls in the checkerboard phase present at
half-filling. The phases are characterised at zero temperatures using numerical
simulations, and mean field theory is used to construct a finite temperature
phase diagram.Comment: 8 pages, 8 figure
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