184,728 research outputs found
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 -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
Synchronisation Properties of Trees in the Kuramoto Model
We consider the Kuramoto model of coupled oscillators, specifically the case
of tree networks, for which we prove a simple closed-form expression for the
critical coupling. For several classes of tree, and for both uniform and
Gaussian vertex frequency distributions, we provide tight closed form bounds
and empirical expressions for the expected value of the critical coupling. We
also provide several bounds on the expected value of the critical coupling for
all trees. Finally, we show that for a given set of vertex frequencies, there
is a rearrangement of oscillator frequencies for which the critical coupling is
bounded by the spread of frequencies.Comment: 21 pages, 19 Figure
Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem
We consider the numbers arising in the problem of normal ordering of
expressions in canonical boson creation and annihilation operators. We treat a
general form of a boson string which is shown to be associated with
generalizations of Stirling and Bell numbers. The recurrence relations and
closed-form expressions (Dobiski-type formulas) are obtained for these
quantities by both algebraic and combinatorial methods. By extensive use of
methods of combinatorial analysis we prove the equivalence of the
aforementioned problem to the enumeration of special families of graphs. This
link provides a combinatorial interpretation of the numbers arising in this
normal ordering problem.Comment: 10 pages, 5 figure
Resource costs for fault-tolerant linear optical quantum computing
Linear optical quantum computing (LOQC) seems attractively simple:
information is borne entirely by light and processed by components such as beam
splitters, phase shifters and detectors. However this very simplicity leads to
limitations, such as the lack of deterministic entangling operations, which are
compensated for by using substantial hardware overheads. Here we quantify the
resource costs for full scale LOQC by proposing a specific protocol based on
the surface code. With the caveat that our protocol can be further optimised,
we report that the required number of physical components is at least five
orders of magnitude greater than in comparable matter-based systems. Moreover
the resource requirements grow higher if the per-component photon loss rate is
worse than one in a thousand, or the per-component noise rate is worse than
. We identify the performance of switches in the network as the single
most influential factor influencing resource scaling
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
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