26,398 research outputs found
On colored set partitions of type
Generalizing Reiner's notion of set partitions of type , we define
colored -partitions by coloring the elements in and not in the zero-block
respectively. Considering the generating function of colored -partitions,
we get the exact formulas for the expectation and variance of the number of
non-zero-blocks in a random colored -partition. We find an asymptotic
expression of the total number of colored -partitions up to an error of
, and prove that the centralized and normalized
number of non-zero-blocks is asymptotic normal over colored -partitions.Comment: 10 page
The Real-Rootedness and Log-concavities of Coordinator Polynomials of Weyl Group Lattices
It is well-known that the coordinator polynomials of the classical root
lattice of type and those of type are real-rooted. They can be
obtained, either by the Aissen-Schoenberg-Whitney theorem, or from their
recurrence relations. In this paper, we develop a trigonometric substitution
approach which can be used to establish the real-rootedness of coordinator
polynomials of type . We also find the coordinator polynomials of type
are not real-rooted in general. As a conclusion, we obtain that all
coordinator polynomials of Weyl group lattices are log-concave.Comment: 8 page
Root geometry of polynomial sequences III: Type with positive coefficients
In this paper, we study the root distribution of some univariate polynomials
satisfying a recurrence of order two with linear polynomial
coefficients over positive numbers. We discover a sufficient and necessary
condition for the overall real-rootedness of all the polynomials, in terms of
the polynomial coefficients of the recurrence. Moreover, in the real-rooted
case, we find the set of limits of zeros, which turns out to be the union of a
closed interval and one or two isolated points; when non-real-rooted polynomial
exists, we present a sufficient condition under which every polynomial with
large has a real zero.Comment: 14page, 3 figure
Surface embedding of non-bipartite -extendable graphs
We find the minimum number for any surface , such
that every -embeddable non-bipartite graph is not -extendable. In
particular, we construct the so-called bow-tie graphs , and
show that they are -extendable. This confirms the existence of an infinite
number of -extendable non-bipartite graphs which can be embedded in the
Klein bottle.Comment: 17 pages, 5 figure
A Note on -Factors of Regular Graphs
Let be an odd integer, and an even integer. In this note, we present
-regular graphs which have no -factors for all . This gives a negative answer to a problem posed by Akbari and
Kano recently
The maximum number of perfect matchings of semi-regular graphs
Let be an even integer, and . In this
paper, we prove that every -graph of order contains
disjoint perfect matchings. This result is sharp in the
sense that (i) there exists a -graph containing exactly
disjoint perfect matchings, and that (ii) there exists a
-graph without perfect matchings for each . As a
consequence, for any integer , every -graph of order
contains disjoint perfect matchings. This extends Csaba
et~al.'s breathe-taking result that every -regular graph of sufficiently
large order is -factorizable, generalizes Zhang and Zhu's result that every
-regular graph of order contains disjoint perfect
matchings, and improves Hou's result that for all , every
-graph of order contains
disjoint perfect matchings.Comment: 30 pages, 9 figure
Piecewise interlacing zeros of polynomials
We introduce the concept of piecewise interlacing zeros for studying the
relation of root distribution of two polynomials. The concept is pregnant with
an idea of confirming the real-rootedness of polynomials in a sequence. Roughly
speaking, one constructs a collection of disjoint intervals such that one may
show by induction that consecutive polynomials have interlacing zeros over each
of the intervals. We confirm the real-rootedness of some polynomials satisfying
a recurrence with linear polynomial coefficients. This extends Gross et al.'s
work where one of the polynomial coefficients is a constant.Comment: 18 pages, 6 figure
The Tutte's condition in terms of graph factors
Let be a connected general graph of even order, with a function . We obtain that satisfies the Tutte's condition with
respect to if and only if contains an -factor for any function
such that for each , where the set consists of the integer and all positive
odd integers less than , and the set consists of positive odd
integers less than or equal to . We also obtain a characterization for
graphs of odd order satisfying the Tutte's condition with respect to a
function.Comment: 5 page
Common zeros of polynomials satisfying a recurrence of order two
We give a characterization of common zeros of a sequence of univariate
polynomials defined by a recurrence of order two with polynomial
coefficients, and with . Real common zeros for such polynomials with
real coefficients are studied further. This paper contributes to the study of
root distribution of recursive polynomial sequences.Comment: 12 page
Counting subwords in flattened permutations
In this paper, we consider the number of occurrences of descents, ascents,
123-subwords, 321-subwords, peaks and valleys in flattened permutations, which
were recently introduced by Callan in his study of finite set partitions. For
descents and ascents, we make use of the kernel method and obtain an explicit
formula (in terms of Eulerian polynomials) for the distribution on
in the flattened sense. For the other four patterns in
question, we develop a unified approach to obtain explicit formulas for the
comparable distributions. We find that the formulas so obtained for 123- and
321-subwords can be expressed in terms of the Chebyshev polynomials of the
second kind, while those for peaks and valleys are more related to the Eulerian
polynomials. We also provide a bijection showing the equidistribution of
descents in flattened permutations of a given length with big descents in
permutations of the same length in the usual sense.Comment: 21 page
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