166 research outputs found
Recurrence relations for patterns of type in flattened permutations
We consider the problem of counting the occurrences of patterns of the form
within flattened permutations of a given length. Using symmetric
functions, we find recurrence relations satisfied by the distributions on
for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a
unified approach to obtain explicit formulas. By these recurrences, we are able
to determine simple closed form expressions for the number of permutations
that, when flattened, avoid one of these patterns as well as expressions for
the average number of occurrences. In particular, we find that the average
number of 23-1 patterns and the average number of 32-1 patterns in
, taken over all permutations of the same length,
are equal, as are the number of permutations avoiding either of these patterns.
We also find that the average number of 21-3 patterns in
over all is the same as it is for 31-2 patterns.Comment: 19 pages. Final version will be published in Journal of Difference
Equations and Application
Relative locations of subwords in free operated semigroups and Motzkin words
Bracketed words are basic structures both in mathematics (such as Rota-Baxter
algebras) and mathematical physics (such as rooted trees) where the locations
of the substructures are important. In this paper we give the classification of
the relative locations of two bracketed subwords of a bracketed word in an
operated semigroup into the separated, nested and intersecting cases. We
achieve this by establishing a correspondence between relative locations of
bracketed words and those of words by applying the concept of Motzkin words
which are the algebraic forms of Motzkin paths.Comment: 14 page
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
Hopf algebras of endomorphisms of Hopf algebras
In the last decennia two generalizations of the Hopf algebra of symmetric
functions have appeared and shown themselves important, the Hopf algebra of
noncommutative symmetric functions NSymm and the Hopf algebra of quasisymmetric
functions QSymm. It has also become clear that it is important to understand
the noncommutative versions of such important structures as Symm the Hopf
algebra of symmetric functions. Not least because the right noncommmutative
versions are often more beautiful than the commutaive ones (not all cluttered
up with counting coefficients). NSymm and QSymm are not truly the full
noncommutative generalizations. One is maximally noncommutative but
cocommutative, the other is maximally non cocommutative but commutative. There
is a common, selfdual generalization, the Hopf algebra of permutations of
Malvenuto, Poirier, and Reutenauer (MPR). This one is, I feel, best understood
as a Hopf algebra of endomorphisms. In any case, this point of view suggests
vast generalizations leading to the Hopf algebras of endomorphisms and word
Hopf algebras with which this paper is concerned. This point of view also sheds
light on the somewhat mysterious formulas of MPR and on the question where all
the extra structure (such as autoduality) comes from. The paper concludes with
a few sections on the structure of MPR and the question of algebra retractions
of the natural inclusion of Hopf algebras of NSymm into MPR and section of the
naural projection of MPR onto QSymm.Comment: 40 pages. Revised and expanded version of a (nonarchived) preprint of
200
k-Spectra of weakly-c-Balanced Words
A word is a scattered factor of if can be obtained from by
deleting some of its letters. That is, there exist the (potentially empty)
words , and such that and
. We consider the set of length- scattered
factors of a given word w, called here -spectrum and denoted
\ScatFact_k(w). We prove a series of properties of the sets \ScatFact_k(w)
for binary strictly balanced and, respectively, -balanced words , i.e.,
words over a two-letter alphabet where the number of occurrences of each letter
is the same, or, respectively, one letter has -more occurrences than the
other. In particular, we consider the question which cardinalities n=
|\ScatFact_k(w)| are obtainable, for a positive integer , when is
either a strictly balanced binary word of length , or a -balanced binary
word of length . We also consider the problem of reconstructing words
from their -spectra
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