Noncrossing Linked Partitions and Large (3,2)-Motzkin Paths

Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and colored Motzkin paths. A (3,2)-Motzkin path can be viewed as a colored Motzkin path in the sense that there are three types of level steps and two types of down steps. A large (3,2)-Motzkin path is defined to be a (3,2)-Motzkin path for which there are only two types of level steps on the x-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of [n+1] and the set of large (3,2)-Motzkin paths of length n. In this setting, we get a simple explanation of the well-known relation between the large and the little Schroder numbers.Comment: 8 page

The Limiting Distribution of the Coefficients of the qq-Catalan Numbers

We show that the limiting distributions of the coefficients of the $q$-Catalan numbers and the generalized $q$-Catalan numbers are normal. Despite the fact that these coefficients are not unimodal for small $n$, we conjecture that for sufficiently large $n$, the coefficients are unimodal and even log-concave except for a few terms of the head and tail.Comment: 13 page

On Balanced Colorings of the n-Cube

A 2-coloring of the n-cube in the n-dimensional Euclidean space can be considered as an assignment of weights of 1 or 0 to the vertices. Such a colored n-cube is said to be balanced if its center of mass coincides with its geometric center. Let $B_{n,2k}$ be the number of balanced 2-colorings of the n-cube with 2k vertices having weight 1. Palmer, Read and Robinson conjectured that for $n\geq 1$, the sequence $\{B_{n,2k}\}_{k=0, 1 ... 2^{n-1}}$ is symmetric and unimodal. We give a proof of this conjecture. We also propose a conjecture on the log-concavity of $B_{n,2k}$ for fixed k, and by probabilistic method we show that it holds when n is sufficiently large.Comment: 10 page

Balanced Properties of the q-Derangement Numbers and the q-Catalan Numbers

Based on Bona's condition for the balanced property of the number of cycles of permutations, we give a general criterion for the balanced property in terms of the generating function of a statistic. We show that the q-derangement numbers and the q-Catalan numbers satisfy the balanced property.Comment: 9 page

Linked Partitions and Permutation Tableaux

Linked partitions are introduced by Dykema in the study of transforms in free probability theory, whereas permutation tableaux are introduced by Steingr\'{i}msson and Williams in the study of totally positive Grassmannian cells. Let $[n]=\{1,2,\ldots,n\}$. Let $L(n,k)$ denote the set of linked partitions of $[n]$ with $k$ blocks, let $P(n,k)$ denote the set of permutations of $[n]$ with $k$ descents, and let $T(n,k)$ denote the set of permutation tableaux of length $n$ with $k$ rows. Steingr\'{i}msson and Williams found a bijection between the set of permutation tableaux of length $n$ with $k$ rows and the set of permutations of $[n]$ with $k$ weak excedances. Corteel and Nadeau gave a bijection from the set of permutation tableaux of length $n$ with $k$ columns to the set of permutations of $[n]$ with $k$ descents. In this paper, we establish a bijection between $L(n,k)$ and $P(n,k-1)$ and a bijection between $L(n,k)$ and $T(n,k)$. Restricting the latter bijection to noncrossing linked partitions, we find that the corresponding permutation tableaux can be characterized by pattern avoidance.Comment: 11 pages, 9 figure

Families of Sets with Intersecting Clusters

A family of $k$-subsets $A_1, A_2, ..., A_d$ on $[n]=\{1,2,..., n\}$ is called a $(d, c)$-cluster if the union $A_1\cup A_2 \cup ... \cup A_d$ contains at most $ck$ elements with $c<d$. Let $\mathcal{F}$ be a family of $k$-subsets of an $n$-element set. We show that for $k \geq 2$ and $n \geq k+2$, if every $(k, 2)$-cluster of $\mathcal{F}$ is intersecting, then $\mathcal{F}$ contains no $(k-1)$-dimensional simplices. This leads to an affirmative answer to Mubayi's conjecture for $d=k$ based on Chv\'atal's simplex theorem. We also show that for any $d$ satisfying $3 \leq d \leq k$ and $n \geq \frac{dk}{d-1}$, if every $(d, {d+1\over 2})$-cluster is intersecting, then $|\mathcal{F}|\leq {{n-1} \choose {k-1}}$ with equality only when $\mathcal{F}$ is a complete star. This result is an extension of both Frankl's theorem and Mubayi's theorem.Comment: 14 pages; Final version, to appear in SIAM J. Discrete Mat

Partitions of Zn\mathbb{Z}_n into Arithmetic Progressions

We introduce the notion of arithmetic progression blocks or AP-blocks of $\mathbb{Z}_n$, which can be represented as sequences of the form $(x, x+m, x+2m, ..., x+(i-1)m) \pmod n$. Then we consider the problem of partitioning $\mathbb{Z}_n$ into AP-blocks for a given difference $m$. We show that subject to a technical condition, the number of partitions of $\mathbb{Z}_n$ into $m$-AP-blocks of a given type is independent of $m$. When we restrict our attention to blocks of sizes one or two, we are led to a combinatorial interpretation of a formula recently derived by Mansour and Sun as a generalization of the Kaplansky numbers. These numbers have also occurred as the coefficients in Waring's formula for symmetric functions.Comment: 11 pages, 2 figure

Zeta Functions and the Log-behavior of Combinatorial Sequences

In this paper, we use the Riemann zeta function $\zeta(x)$ and the Bessel zeta function $\zeta_{\mu}(x)$ to study the log-behavior of combinatorial sequences. We prove that $\zeta(x)$ is log-convex for $x>1$. As a consequence, we deduce that the sequence $\{|B_{2n}|/(2n)!\}_{n\geq 1}$ is log-convex, where $B_n$ is the $n$-th Bernoulli number. We introduce the function $\theta(x)=(2\zeta(x)\Gamma(x+1))^{\frac{1}{x}}$, where $\Gamma(x)$ is the gamma function, and we show that $\log \theta(x)$ is strictly increasing for $x\geq 6$. This confirms a conjecture of Sun stating that the sequence $\{\sqrt[n] {|B_{2n}}|\}_{n\geq 1}$ is strictly increasing. Amdeberhan, Moll and Vignat defined the numbers $a_n(\mu)=2^{2n+1}(n+1)!(\mu+1)_n\zeta_{\mu}(2n)$ and conjectured that the sequence $\{a_n(\mu)\}_{n\geq 1}$ is log-convex for $\mu=0$ and $\mu=1$. By proving that $\zeta_{\mu}(x)$ is log-convex for $x>1$ and $\mu>-1$, we show that the sequence $\{a_n(\mu)\}_{n\geq 1}$ is log-convex for any $\mu>-1$. We introduce another function $\theta_{\mu}(x)$ involving $\zeta_{\mu}(x)$ and the gamma function $\Gamma(x)$ and we show that $\log \theta_{\mu}(x)$ is strictly increasing for $x>8e(\mu+2)^2$. This implies that $\sqrt[n]{a_n(\mu)} 4e(\mu+2)^2$. Based on Dobinski's formula, we prove that $\sqrt[n]{B_n}<\sqrt[n+1]{B_{n+1}}$ for $n\geq 1$, where $B_n$ is the $n$-th Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of $\{\sqrt[n]{B_n}\}_{n\geq 1}$ and H\"{o}lder's inequality in probability theory.Comment: 16 pages; to appear in Proc. Edinburgh Math. Soc. (2

Schur Positivity and the qq-Log-convexity of the Narayana Polynomials

Using Schur positivity and the principal specialization of Schur functions, we provide a proof of a recent conjecture of Liu and Wang on the $q$-log-convexity of the Narayana polynomials, and a proof of the second conjecture that the Narayana transformation preserves the log-convexity. Based on a formula of Br\"and$\mathrm{\acute{e}}$n which expresses the $q$-Narayana numbers as the specializations of Schur functions, we derive several symmetric function identities using the Littlewood-Richardson rule for the product of Schur functions, and obtain the strong $q$-log-convexity of the Narayana polynomials and the strong $q$-log-concavity of the $q$-Narayana numbers.Comment: 38 pages, 6 figure

Infinitely Log-monotonic Combinatorial Sequences

We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence $\{a_n\}_{n\geq 0}$ is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence $\{a_{n+1}/a_{n}\}_{n\geq 0}$ is log-concave. Furthermore, we prove that if a sequence $\{a_n\}_{n\geq k}$ is ratio log-concave, then the sequence $\{\sqrt[n]{a_n}\}_{n\geq k}$ is strictly log-concave subject to a certain initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers $D_n$, we confirm a conjecture of Sun on the log-concavity of the sequence $\{\sqrt[n]{D_n}\}_{n\geq 1}$.Comment: 26 pages, to appear in Adv. Appl. Mat