99,630 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
Avoidance of Partitions of a Three-element Set
Klazar defined and studied a notion of pattern avoidance for set partitions,
which is an analogue of pattern avoidance for permutations. Sagan considered
partitions which avoid a single partition of three elements. We enumerate
partitions which avoid any family of partitions of a 3-element set as was done
by Simion and Schmidt for permutations. We also consider even and odd set
partitions. We provide enumerative results for set partitions restricted by
generalized set partition patterns, which are an analogue of the generalized
permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit
of work done by Babson and Steingr{'{\i}}msson, we will show how these
generalized partition patterns can be used to describe set partition
statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied
Mathematic
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
Governing Singularities of Schubert Varieties
We present a combinatorial and computational commutative algebra methodology
for studying singularities of Schubert varieties of flag manifolds.
We define the combinatorial notion of *interval pattern avoidance*. For
"reasonable" invariants P of singularities, we geometrically prove that this
governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties
are globally not P. The prototypical case is P="singular"; classical pattern
avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is
insufficient in general.
Our approach is analyzed for some common invariants, including
Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness,
extending [Woo-Yong'05]; the description of the singular locus (which was
independently proved by [Billey-Warrington '03], [Cortez '03],
[Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted.
Our methods are amenable to computer experimentation, based on computing with
*Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using
Macaulay 2. This feature is supplemented by a collection of open problems and
conjectures.Comment: 23 pages. Software available at the authors' webpages. Version 2 is
the submitted version. It has a nomenclature change: "Bruhat-restricted
pattern avoidance" is renamed "interval pattern avoidance"; the introduction
has been reorganize
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
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