3,049 research outputs found

### 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

### Generalized pattern avoidance with additional restrictions

Babson and Steingr\'{\i}msson introduced generalized permutation patterns
that allow the requirement that two adjacent letters in a pattern must be
adjacent in the permutation. We consider n-permutations that avoid the
generalized pattern 1-32 and whose k rightmost letters form an increasing
subword. The number of such permutations is a linear combination of Bell
numbers. We find a bijection between these permutations and all partitions of
an $(n-1)$-element set with one subset marked that satisfy certain additional
conditions. Also we find the e.g.f. for the number of permutations that avoid a
generalized 3-pattern with no dashes and whose k leftmost or k rightmost
letters form either an increasing or decreasing subword. Moreover, we find a
bijection between n-permutations that avoid the pattern 132 and begin with the
pattern 12 and increasing rooted trimmed trees with n+1 nodes.Comment: 18 page

### On graphs with representation number 3

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the
alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if
$(x,y)$ is an edge in $E$. A graph is word-representable if and only if it is
$k$-word-representable for some $k$, that is, if there exists a word containing
$k$ copies of each letter that represents the graph. Also, being
$k$-word-representable implies being $(k+1)$-word-representable. The minimum
$k$ such that a word-representable graph is $k$-word-representable, is called
graph's representation number.
Graphs with representation number 1 are complete graphs, while graphs with
representation number 2 are circle graphs. The only fact known before this
paper on the class of graphs with representation number 3, denoted by
$\mathcal{R}_3$, is that the Petersen graph and triangular prism belong to this
class. In this paper, we show that any prism belongs to $\mathcal{R}_3$, and
that two particular operations of extending graphs preserve the property of
being in $\mathcal{R}_3$. Further, we show that $\mathcal{R}_3$ is not included
in a class of $c$-colorable graphs for a constant $c$. To this end, we extend
three known results related to operations on graphs.
We also show that ladder graphs used in the study of prisms are
$2$-word-representable, and thus each ladder graph is a circle graph. Finally,
we discuss $k$-word-representing comparability graphs via consideration of
crown graphs, where we state some problems for further research

### The sigma-sequence and counting occurrences of some patterns, subsequences and subwords

We consider sigma-words, which are words used by Evdokimov in the
construction of the sigma-sequence. We then find the number of occurrences of
certain patterns and subwords in these words.Comment: 10 page

### Quantum measurements and the Abelian Stabilizer Problem

We present a polynomial quantum algorithm for the Abelian stabilizer problem
which includes both factoring and the discrete logarithm. Thus we extend famous
Shor's results. Our method is based on a procedure for measuring an eigenvalue
of a unitary operator. Another application of this procedure is a polynomial
quantum Fourier transform algorithm for an arbitrary finite Abelian group. The
paper also contains a rather detailed introduction to the theory of quantum
computation.Comment: 22 pages, LATE

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