22,625 research outputs found
Approximation hardness of Shortest Common Superstring variants
The shortest common superstring (SCS) problem has been studied at great
length because of its connections to the de novo assembly problem in
computational genomics. The base problem is APX-complete, but several
generalizations of the problem have also been studied. In particular, previous
results include that SCS with Negative strings (SCSN) is in Log-APX (though
there is no known hardness result) and SCS with Wildcards (SCSW) is
Poly-APX-hard. Here, we prove two new hardness results: (1) SCSN is
Log-APX-hard (and therefore Log-APX-complete) by a reduction from Minimum Set
Cover and (2) SCS with Negative strings and Wildcards (SCSNW) is NPOPB-hard by
a reduction from Minimum Ones 3SAT.Comment: 10 page
Enumeration of Rota-Baxter Words
In this paper, we prove results on enumerations of sets of Rota-Baxter words
in a finite number of generators and a finite number of unary operators.
Rota-Baxter words are words formed by concatenating generators and images of
words under Rota-Baxter operators. Under suitable conditions, they form
canonical bases of free Rota-Baxter algebras and are studied recently in
relation to combinatorics, number theory, renormalization in quantum field
theory, and operads. Enumeration of a basis is often a first step to choosing a
data representation in implementation. Our method applies some simple ideas
from formal languages and compositions (ordered partitions) of an integer. We
first settle the case of one generator and one operator where both have
exponent 1 (the idempotent case). Some integer sequences related to these sets
of Rota-Baxter words are known and connected to other combinatorial sequences,
such as the Catalan numbers, and others are new. The recurrences satisfied by
the generating series of these sequences prompt us to discover an efficient
algorithm to enumerate the canonical basis of certain free Rota-Baxter
algebras. More general sets of Rota-Baxter words are enumerated with summation
techniques related to compositions of integers.Comment: 39 pages; corrected minor errors, improved presentation and
strengthened results; corrected minor errors in Eq (52) and Corol. 5.
Derangements and Relative Derangements of Type
By introducing the notion of relative derangements of type , also called
signed relative derangements, which are defined in terms of signed
permutations, we obtain a type analogue of the well-known relation between
relative derangements and the classical derangements. While this fact can be
proved by using the principle of inclusion and exclusion, we present a
combinatorial interpretation with the aid of the intermediate structure of
signed skew derangements.Comment: 7 page
k-Marked Dyson Symbols and Congruences for Moments of Cranks
By introducing -marked Durfee symbols, Andrews found a combinatorial
interpretation of -th symmetrized moment of ranks of
partitions of . Recently, Garvan introduced the -th symmetrized moment
of cranks of partitions of in the study of the higher-order
spt-function . In this paper, we give a combinatorial interpretation
of . We introduce -marked Dyson symbols based on a
representation of ordinary partitions given by Dyson, and we show that
equals the number of -marked Dyson symbols of . We then
introduce the full crank of a -marked Dyson symbol and show that there exist
an infinite family of congruences for the full crank function of -marked
Dyson symbols which implies that for fixed prime and positive
integers and , there exist infinitely many non-nested
arithmetic progressions such that .Comment: 19 pages, 2 figure
On the Positive Moments of Ranks of Partitions
By introducing -marked Durfee symbols, Andrews found a combinatorial
interpretation of -th symmetrized moment of ranks of
partitions of in terms of -marked Durfee symbols of . In this
paper, we consider the -th symmetrized positive moment of
ranks of partitions of which is defined as the truncated sum over positive
ranks of partitions of . As combintorial interpretations of
and , we show that for fixed and
with , equals the number of
-marked Durfee symbols of with the -th rank being zero and
equals the number of -marked Durfee symbols of
with the -th rank being positive. The interpretations of
and also imply the interpretation
of given by Andrews since equals
plus twice of . Moreover, we obtain
the generating functions of and .Comment: 10 page
Partitions and Partial Matchings Avoiding Neighbor Patterns
We obtain the generating functions for partial matchings avoiding neighbor
alignments and for partial matchings avoiding neighbor alignments and left
nestings. We show that there is a bijection between partial matchings avoiding
three neighbor patterns (neighbor alignments, left nestings and right nestings)
and set partitions avoiding right nestings via an intermediate structure of
integer compositions. Such integer compositions are known to be in one-to-one
correspondence with self-modified ascent sequences or
-avoiding permutations, as shown by Bousquet-M\'elou,
Claesson, Dukes and Kitaev.Comment: 15 pages, 8 figure
Identities Derived from Noncrossing Partitions of Type B
Based on weighted noncrossing partitions of type B, we obtain type B
analogues of Coker's identities on the Narayana polynomials. A parity reversing
involution is given for the alternating sum of Narayana numbers of type B.
Moreover, we find type B analogues of the refinements of Coker's identities due
to Chen, Deutsch and Elizalde. By combinatorial constructions, we provide type
B analogues of three identities of Mansour and Sun also on the Narayana
polynomials.Comment: 18 pages, 4 figure
The Generating Function for the Dirichlet Series
The Dirichlet series are of fundamental importance in number theory.
Shanks defined the generalized Euler and class numbers in connection with these
Dirichlet series, denoted by . We obtain a formula for
the exponential generating function of , where m is an
arbitrary positive integer. In particular, for m>1, say, , where b is
square-free and u>1, we prove that can be expressed as a linear
combination of the four functions , where p is an integer satisfying , and
with being a constant depending on b. Moreover, the
Dirichlet series can be easily computed from the generating function
formula for . Finally, we show that the main ingredient in the formula
for has a combinatorial interpretation in terms of the m-signed
permutations defined by Ehrenborg and Readdy. In principle, this answers a
question posed by Shanks concerning a combinatorial interpretation for the
numbers .Comment: 18 page
Linked Partitions and Linked Cycles
The notion of noncrossing linked partition arose from the study of certain
transforms in free probability theory. It is known that the number of
noncrossing linked partitions of [n+1] is equal to the n-th large Schroder
number , which counts the number of Schroder paths. In this paper we give
a bijective proof of this result. Then we introduce the structures of linked
partitions and linked cycles. We present various combinatorial properties of
noncrossing linked partitions, linked partitions, and linked cycles, and
connect them to other combinatorial structures and results, including
increasing trees, partial matchings, k-Stirling numbers of the second kind, and
the symmetry between crossings and nestings over certain linear graphs.Comment: 22 pages, 11 figure
Labeled Partitions and the q-Derangement Numbers
By a re-examination of MacMahon's original proof of his celebrated theorem on
the distribution of the major indices over permutations, we give a
reformulation of his argument in terms of the structure of labeled partitions.
In this framework, we are able to establish a decomposition theorem for labeled
partitions that leads to a simple bijective proof of Wachs' formula on the
q-derangement numbers.Comment: 6 page
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