This paper is devoted to the study of the log-convexity of combinatorial sequences. We show that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. We develop techniques for dealing with the log-convexity of sequences satisfying a three-term recurrence. We also introduce the concept of $q$-log-convexity and establish the connection with linear transformations preserving the log-convexity. As applications of our results, we prove the log-convexity and $q$-log-convexity of many famous combinatorial sequences of numbers and polynomials.
We give a proof to the following theorem, which is well-known among experts: A connected subcomplex $W$ of a finite dimensional CAT(0) cubed complex $X$ is convex if and only if Lk$(v, W)$ is a full subcomplex of Lk$(v, X)$ for every vertex $v$ of $W$.
In this paper, we prove the $q$-log-convexity of Domb's polynomials, which was conjectured by Sun in the study of Ramanujan-Sato type series for powers of $π$. As a result, we obtain the log-convexity of Domb's numbers. Our proof is based on the $q$-log-convexity of Narayana polynomials of type $B$ and a criterion for determining $q$-log-convexity of self-reciprocal polynomials.
Let $P_{n,k}$ be the number of permutations $π$ on [n]={1, 2,..., n} such that the length of the longest increasing subsequences of $π$ equals k, and let $M_{2n, k}$ be the number of matchings on [2n] with crossing number k. Define $P_n(x)= \sum_k P_{n,k}x^k$ and $M_{2n}(x)=\sum_{k} M_{2n,k}x^k$. We propose some conjectures on the log-concavity and q-log-convexity of the polynomials $P_n(x)$ and $M_{2n}(x)$. We also introduce the notions of $\infty$-q-log-convexity and $\infty$-q-log-concavity, and the notion of higher order log-concavity with respect to $\infty$-q-log-convex or $\infty$-q-log…
A sequence $\{ a_n \}_{n \ge 0}$ is said to be asymptotically $r$-log-convex if it is $r$-log-convex for $n$ sufficiently large. We present a criterion on the asymptotical $r$-log-convexity based on the asymptotic behavior of $a_n a_{n+2}/a_{n+1}^2$. As an application, we show that most P-recursive sequences are asymptotic $r$-log-convexity for any integer $r$ once they are log-convex. Moreover, for a concrete integer $r$, we present a systematic method to find the explicit integer $N$ such that a P-recursive sequence $\{a_n\}_{n \ge N}$ is $r$-log-convex. This enable us to prove the $r$-log-c…
In this paper we prove the strong $q$-log-convexity of the Eulerian polynomials of Coxeter groups using their exponential generating functions. Our proof is based on the theory of exponential Riordan arraya and a criterion for determining the strong $q$-log-convexity of polynomials sequences, whose generating functions can be given by the continued fraction. As consequences, we get the strong $q$-log-convexity the Eulerian polynomials of type $A_n,B_n$, their $q$-analogous and the generalized Eulerian polynomials associated to the arithmetic progression $\{a,a+d,a+2d,a+3d,\ldots\}$ in a unifie…
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 Bränd$\mathrm{\acute{e}}$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 pol…
Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and that Motzkin numbers and secondary structure numbers of rank 1 are log-convex. In fact, we prove via calculus a much stronger result that a natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of Motzkin numbers and secondary structures recursions are increasing functions. We indicate how to prove asymptotically the log-convexity for general…
The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q=1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted.
We prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence $\{\sum_{k=0}^n{n\choose k}^2q^k\}_{n\geq 0}$. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular arra…
We study a combinatorial notion where given a set of lattice points one takes the set of all sums of subsets of a fixed size, and we ask if the given set comes from a convex lattice polytope whether the resulting set also comes from a convex lattice polytope. We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.
The Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence $\{P^2_n-P_{n-1}P_{n+1}\}_{n\geq 1}$ are strictly log-convex and hence the Catalan-Larcombe-French sequence is strictly 2-log-convex.
In his study of Ramanujan-Sato type series for $1/π$, Sun introduced a sequence of polynomials $S_n(q)$ as given by $$S_n(q)=\sum\limits_{k=0}^n{n\choose k}{2k\choose k}{2(n-k)\choose n-k}q^k,$$ and he conjectured that the polynomials $S_n(q)$ are $q$-log-convex. By imitating a result of Liu and Wang on generating new $q$-log-convex sequences of polynomials from old ones, we obtain a sufficient condition for determining the $q$-log-convexity of self-reciprocal polynomials. Based on this criterion, we then give an affirmative answer to Sun's conjecture.
Tverberg's theorem asserts that every (k-1)(d+1)+1 points in R^d can be partitioned into k parts, so that the convex hulls of the parts have a common intersection. Calder and Eckhoff asked whether there is a purely combinatorial deduction of Tverberg's theorem from the special case k=2. We dash the hopes of a purely combinatorial deduction, but show that the case k=2 does imply that every set of O(k^2 log^2 k) points admits a Tverberg partition into k parts.
We extend the exponential formula by Bender and Canfield (1996), which relates log-concavity and the cycle index polynomials. The extension clarifies the log-convexity relation. The proof is by noticing the property of a compound Poisson distribution together with its moment generating function. We also give a combinatorial proof of extended "log-convex part" referring Bender and Canfield's approach, where the formula by Bruijn and Erdös (1953) is additionally exploited. The combinatorial approach yields richer structural results more than log-convexity. Furthermore, we consider normal and bin…
In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $$A_{n,k}(q)=A_{n-1,k-1}(q)+g_k(q)A_{n-1,k}(q)+h_{k+1}(q)A_{n-1,k+1}(q).$$ Those criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the $2$-$q$-log-convexity of alternating Eulerian polynomials, $2$-log-convexity of Eu…
We define a notion of morphism between combinatorial codes, making the class of all combinatorial codes into a category $\mathbf{Code}$. We show that morphisms can be used to remove redundant information from a code, and that morphisms preserve convexity. This fact leads us to define "minimally non-convex" codes. We propose a program to characterize these minimal obstructions to convexity and hence characterize all convex codes. We implement a library of Sage code to perform computation with morphisms. These computational methods yield the smallest to-date example of a non-convex code with no …
In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In particular, we show that for any finite set $A\subset{\mathbb{R}}$ and any strictly convex or concave function $f$, \[|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}\] and \[\max\{|A-A|,\ |f(A)+f(A)|\}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}.\] For the latter of these inequalities, we go on to consider the consequences for a sum-product type problem.
Let $\{b_{k}(n)\}_{n=0}^{\infty}$ be the Bell numbers of order $k$. It is proved that the sequence $\{b_{k}(n)/n!\}_{n=0}^{\infty}$ is log-concave and the sequence $\{b_{k}(n)\}_{n=0}^{\infty}$ is log-convex, or equivalently, the following inequalities hold for all $n\geq 0$, $$1\leq {b_{k}(n+2) b_{k}(n) \over b_{k}(n+1)^{2}} \leq {n+2 \over n+1}.$$ Let $\{\a(n)\}_{n=0}^{\infty}$ be a sequence of positive numbers with $\a(0)=1$. We show that if $\{\a(n)\}_{n=0}^{\infty}$ is log-convex, then $$\a (n) \a (m) \leq \a(n+m), \quad \forall n, m\geq 0.$$ On the other hand, if $\{\a(n)/n!\}_{n=0}^{\in…
We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).
We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apery numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4 and the large Schroder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are k-log-convex for any $k\geq 1$ possibly except for a constant number of terms at the beginning.
Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of parameter. Four types of such power series are considered where parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of $q$-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions…
We combinatorially prove that the number $R(n,k)$ of permutations of length $n$ having $k$ runs is a log-concave sequence in $k$, for all $n$. We also give a new combinatorial proof for the log-concavity of the Eulerian numbers.
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial ideals, subdivisions of products of simplices, matroid theory, finite metric spaces, and the tropical Grassmannians. The relationship between these topics is explained via one running example throughout the whole paper. The final section explains how the new version 2.9.4 of the software system polymake can be used to compute with tropical polytopes.
Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals, which are called the Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ and the Fennessey-Larcombe-French sequence $\{V_n\}_{n\geq 0}$ respectively. In this paper, we prove the log-convexity of $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$ and $\{n!V_n\}_{n\geq 1}$, the ratio log-concavity of $\{P_n\}_{n\geq 0}$ and the sequence $\{A_n\}_{n\geq 0}$ of Apéry numbers, and the ratio log-convexity of $\{V_n\}_{n\geq 1}$.
The purpose of this note is to explain that the combinatorial local log-concavity conjecture introduced by Gross, Mansour, Tucker and Wang (Eur. J. Comb. 52, 207-222, 2016) in fact follows from a result of Stanley (Eur. J. Comb. 32 (6), 937-943, 2011).
It is well-known that the binomial transformation preserves the log-concavity property and log-convexity property. Let $\binom{a+n}{b+k}$ be the binomial coefficients and $\binom{n,k}{j}$ be defined by $(b_0+b_1x+\cdots+b_kx^{k})^n:=\sum_{j=0}^{kn}\binom{n,k}{j}x^j,$ where the sequence $(b_i)_{0\leq i\leq k}$ is log-concave. In this paper, we prove that the linear transformation $$y_n(q)=\sum_{k=0}^n\binom{a+n}{b+k}x_k(q)$$ preserves the strong $q$-log-concavity property for any fixed nonnegative integers $a$ and $b$, which strengthens and gives a simple proof of results of Ehrenborg and Stein…
A sequence $\{z_n\}_{n\geq0}$ is called ratio log-convex in the sense that the ratio sequence $\{\frac{z_{n+1}}{z_n}\}_{n\geq0}$ is log-convex. Based on a three-term recurrence for sequences, we develop techniques for dealing with the ratio log-convexity of ratio sequences. As applications, we prove that the ratio sequences of numbers, including the derangement numbers, the Motzkin numbers, the Fine numbers, Franel numbers and the Domb numbers are ratio log-convex, respectively. Finally, we not only prove that the sequence of derangement numbers is asymptotically infinitely log-monotonic, but …
Two general methods for establishing the logarithmic behavior of recursively defined sequences of real numbers are presented. One is the interlacing method, and the other one is based on calculus. Both methods are used to prove logarithmic behavior of some combinatorially relevant sequences, such as Motzkin and Schröder numbers, sequences of values of some classic orthogonal polynomials, and many others. The calculus method extends also to two- (or more-) indexed sequences.
We consider log-convex sequences that satisfy an additional constraint imposed on their rate of growth. We call such sequences log-balanced. It is shown that all such sequences satisfy a pair of double inequalities. Sufficient conditions for log-balancedness are given for the case when the sequence satisfies a two- (or more-) term linear recurrence. It is shown that many combinatorially interesting sequences belong to this class, and, as a consequence, that the above-mentioned double inequalities are valid for all of them.
Friedland (1981) showed that for a nonnegative square matrix A, the spectral radius r(e^D A) is a log-convex functional over the real diagonal matrices D. He showed that for fully indecomposable A, log r(e^D A) is strictly convex over D_1, D_2 if and only if D_1-D_2 != c I for any c \in R. Here the condition of full indecomposability is shown to be replaceable by the weaker condition that A and A'A be irreducible, which is the sharpest possible replacement condition. Irreducibility of both A and A'A is shown to be equivalent to irreducibility of A^2 and A'A, which is the condition for a number…
We prove a new class of low-energy decompositions which, amongst other consequences, imply that any finite set $A$ of integers may be written as $A = B \cup C$, where $B$ and $C$ are disjoint sets satisfying \[ |\{ (b_1, \dots, b_{2s}) \in B^{2s} \ | \ b_1 + \dots + b_{s} = b_{s+1} + \dots + b_{2s}\}| \ll_{s} |B|^{2s - (\log \log s)^{1/2 - o(1)}} \] and \[ |\{ (c_1, \dots, c_{2s}) \in C^{2s} \ | \ c_1 \dots c_{s} = c_{s+1} \dots c_{2s} \}| \ll_{s} |C|^{2s - (\log \log s)^{1/2 - o(1)}}.\] This generalises previous results of Bourgain--Chang on many-fold sumsets and product sets to the setting o…
We study combinatorial Dyson Schwinger equations, expressed in the Hopf algebra of words with a quasi shuffle product. We map them into an algebra of polynomials in one indeterminate L and show that the leading log expansion one obtains with such a mapping are simple power law like expression
Let $p(n)$ denote the partition function. Desalvo and Pak proved the log-concavity of $p(n)$ for $n>25$ and the inequality $\frac{p(n-1)}{p(n)}\left(1+\frac{1}{n}\right)>\frac{p(n)}{p(n+1)}$ for $n>1$. Let $r(n)=\sqrt[n]{p(n)/n}$ and $Δ$ be the difference operator respect to $n$. Desalvo and Pak pointed out that their approach to proving the log-concavity of $p(n)$ may be employed to prove a conjecture of Sun on the log-convexity of $\{r(n)\}_{n\geq 61}$, as long as one finds an appropriate estimate of $Δ^2 \log r(n-1)$. In this paper, we obtain a lower bound for $Δ^2\log r(n-1)$, leading to a…
We investigate a canonical extension of a conventional combinatorial notion of reduced divisors to a notion of tropical projections, which can be defined as the unique minimizers of the so-called $B$-pseudonorms with respect to compact tropical convex sets. In this paper, we build the foundation of a theory of idempotent analysis using tropical projections and obtain a series of subsequent results, e.g. tropical retracts, construction of compact tropical convex sets and a set-theoretical characterization of tropical weak independence. In particular, we prove a tropical version of Mazur's Theor…
In this paper, we introduce the $b$-bibranching problem in digraphs, which is a common generalization of the bibranching and $b$-branching problems. The bibranching problem, introduced by Schrijver (1982), is a common generalization of the branching and bipartite edge cover problems. Previous results on bibranchings include polynomial algorithms, a linear programming formulation with total dual integrality, a packing theorem, and an M-convex submodular flow formulation. The $b$-branching problem, recently introduced by Kakimura, Kamiyama, and Takazawa (2018), is a generalization of the branchi…
We prove a \emph{query complexity} lower bound on rank-one principal component analysis (PCA). We consider an oracle model where, given a symmetric matrix $M \in \mathbb{R}^{d \times d}$, an algorithm is allowed to make $T$ \emph{exact} queries of the form $w^{(i)} = Mv^{(i)}$ for $i \in \{1,\dots,T\}$, where $v^{(i)}$ is drawn from a distribution which depends arbitrarily on the past queries and measurements $\{v^{(j)},w^{(j)}\}_{1 \le j \le i-1}$. We show that for a small constant $ε$, any adaptive, randomized algorithm which can find a unit vector $\widehat{v}$ for which $\widehat{v}^{\top}…
