9,620 research outputs found
On multi-avoidance of generalized patterns
In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no
internal dashes, that is, where the patterns correspond to contiguous subwords
in a permutation. In three essentially different cases, the numbers of such
-permutations are , the number of involutions in ,
and , where is the -th Euler number. In this paper we give
recurrence relations for the remaining three essentially different cases.
To complete the descriptions in [Kit3] and [KitMans], we consider avoidance
of a pattern of the form (a classical 3-pattern) and beginning or
ending with an increasing or decreasing pattern. Moreover, we generalize this
problem: we demand that a permutation must avoid a 3-pattern, begin with a
certain pattern and end with a certain pattern simultaneously. We find the
number of such permutations in case of avoiding an arbitrary generalized
3-pattern and beginning and ending with increasing or decreasing patterns.Comment: 26 page
The problem of the pawns
In this paper we study the number of ways to place nonattacking
pawns on an chessboard. We find an upper bound for and
analyse its asymptotic behavior. It turns out that
exists and is bounded from above by
. Also, we consider a lower bound for by reducing
this problem to that of tiling an board with square tiles
of size and . Moreover, we use the transfer-matrix
method to implement an algorithm that allows us to get an explicit formula for
for given .Comment: 16 pages; 6 figure
The Peano curve and counting occurrences of some patterns
We introduce Peano words, which are words corresponding to finite
approximations of the Peano space filling curve. We then find the number of
occurrences of certain patterns in these words.Comment: 9 pages, 1 figur
Enumeration of 3-letter patterns in compositions
Let A be any set of positive integers and n a positive integer. A composition
of n with parts in A is an ordered collection of one or more elements in A
whose sum is n. We derive generating functions for the number of compositions
of n with m parts in A that have r occurrences of 3-letter patterns formed by
two (adjacent) instances of levels, rises and drops. We also derive asymptotics
for the number of compositions of n that avoid a given pattern. Finally, we
obtain the generating function for the number of k-ary words of length m which
contain a prescribed number of occurrences of a given pattern as a special case
of our results.Comment: 20 pages, 1 figure; accepted for the Proceedings of the 2005 Integer
Conferenc
Variational methods for fractional -Sturm--Liouville Problems
In this paper, we formulate a regular -fractional Sturm--Liouville problem
(qFSLP) which includes the left-sided Riemann--Liouville and the right-sided
Caputo -fractional derivatives of the same order , . We introduce the essential -fractional variational analysis needed
in proving the existence of a countable set of real eigenvalues and associated
orthogonal eigenfunctions for the regular qFSLP when associated
with the boundary condition . A criteria for the first eigenvalue
is proved. Examples are included. These results are a generalization of the
integer regular -Sturm--Liouville problem introduced by Annaby and Mansour
in [1]
Simultaneous avoidance of generalized patterns
In [BabStein] 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. In [Kit1] Kitaev considered
simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no
internal dashes, that is, where the patterns correspond to contiguous subwords
in a permutation. There either an explicit or a recursive formula was given for
all but one case of simultaneous avoidance of more than two patterns.
In this paper we find the exponential generating function for the remaining
case. Also we consider permutations that avoid a pattern of the form or
and begin with one of the patterns , , ,
or end with one of the patterns , ,
, . For each of these cases we find either the
ordinary or exponential generating functions or a precise formula for the
number of such permutations. Besides we generalize some of the obtained results
as well as some of the results given in [Kit3]: we consider permutations
avoiding certain generalized 3-patterns and beginning (ending) with an
arbitrary pattern having either the greatest or the least letter as its
rightmost (leftmost) letter.Comment: 18 page
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