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 BB

By introducing the notion of relative derangements of type $B$, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type $B$ 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 $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $\eta_{2k}(n)$ of ranks of partitions of $n$. Recently, Garvan introduced the $2k$-th symmetrized moment $\mu_{2k}(n)$ of cranks of partitions of $n$ in the study of the higher-order spt-function $spt_k(n)$. In this paper, we give a combinatorial interpretation of $\mu_{2k}(n)$. We introduce $k$-marked Dyson symbols based on a representation of ordinary partitions given by Dyson, and we show that $\mu_{2k}(n)$ equals the number of $(k+1)$-marked Dyson symbols of $n$. We then introduce the full crank of a $k$-marked Dyson symbol and show that there exist an infinite family of congruences for the full crank function of $k$-marked Dyson symbols which implies that for fixed prime $p\geq 5$ and positive integers $r$ and $k\leq (p-1)/2$, there exist infinitely many non-nested arithmetic progressions $An+B$ such that $\mu_{2k}(An+B)\equiv 0\pmod{p^r}$.Comment: 19 pages, 2 figure

On the Positive Moments of Ranks of Partitions

By introducing $k$-marked Durfee symbols, Andrews found a combinatorial interpretation of $2k$-th symmetrized moment $\eta_{2k}(n)$ of ranks of partitions of $n$ in terms of $(k+1)$-marked Durfee symbols of $n$. In this paper, we consider the $k$-th symmetrized positive moment $\bar{\eta}_k(n)$ of ranks of partitions of $n$ which is defined as the truncated sum over positive ranks of partitions of $n$. As combintorial interpretations of $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$, we show that for fixed $k$ and $i$ with $1\leq i\leq k+1$, $\bar{\eta}_{2k-1}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being zero and $\bar{\eta}_{2k}(n)$ equals the number of $(k+1)$-marked Durfee symbols of $n$ with the $i$-th rank being positive. The interpretations of $\bar{\eta}_{2k-1}(n)$ and $\bar{\eta}_{2k}(n)$ also imply the interpretation of $\eta_{2k}(n)$ given by Andrews since $\eta_{2k}(n)$ equals $\bar{\eta}_{2k-1}(n)$ plus twice of $\bar{\eta}_{2k}(n)$. Moreover, we obtain the generating functions of $\bar{\eta}_{2k}(n)$ and $\bar{\eta}_{2k-1}(n)$.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 $3\bar{1}52\bar{4}$-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 Lm(s)L_m(s)

The Dirichlet series $L_m(s)$ are of fundamental importance in number theory. Shanks defined the generalized Euler and class numbers in connection with these Dirichlet series, denoted by $\{s_{m,n}\}_{n\geq 0}$. We obtain a formula for the exponential generating function $s_m(x)$ of $s_{m,n}$, where m is an arbitrary positive integer. In particular, for m>1, say, $m=bu^2$, where b is square-free and u>1, we prove that $s_m(x)$ can be expressed as a linear combination of the four functions $w(b,t)\sec (btx)(\pm \cos ((b-p)tx)\pm \sin (ptx))$, where p is an integer satisfying $0\leq p\leq b$, $t|u^2$ and $w(b,t)=K_bt/u$ with $K_b$ being a constant depending on b. Moreover, the Dirichlet series $L_m(s)$ can be easily computed from the generating function formula for $s_m(x)$. Finally, we show that the main ingredient in the formula for $s_{m,n}$ 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 $s_{m,n}$.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 $r_n$, 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