A triangle $\{a(n,k)\}_{0\le k\le n}$ of nonnegative numbers is LC-positive if for each $r$, the sequence of polynomials $\sum_{k=r}^{n}a(n,k)q^k$ is $q$-log-concave. It is double LC-positive if both triangles $\{a(n,k)\}$ and $\{a(n,n-k)\}$ are LC-positive. We show that if $\{a(n,k)\}$ is LC-positive then the log-concavity of the sequence $\{x_k\}$ implies that of the sequence $\{z_n\}$ defined by $z_n=\sum_{k=0}^{n}a(n,k)x_k$, and if $\{a(n,k)\}$ is double LC-positive then the log-concavity of sequences $\{x_k\}$ and $\{y_k\}$ implies that of the sequence $\{z_n\}$ defined by $z_n=\sum_{k=0}^{n}a(n,k)x_ky_{n-k}$. Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.
We prove Okounkov's conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An alternative proof of this result is provided. We also give an intriguing log-concavity property of Schur functions.
Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen's log-concavity conjecture, Bóna, Lackner and Sagan further studied similar problems by restricting the whole symmetric group to certain of its subsets. They obtained the log-concavity of the corresponding generating functions for these subsets by using the hook-length formula. In this paper, we generalize and prove their results by establishing the Schur positivity of certain symmetric functions. Thi…
Menon's proof of the preservation of log-concavity of sequences under convolution becomes simpler when adapted to 2-sided infinite sequences. Under assumption of log-concavity of two 2-sided infinite sequences, the existence of the convolution is characterised by a convergence criterion. Preservation of log-concavity under convolution yields a method of establishing the log-concavity of rows of a large class of Pascal type triangles, including a weighted generalization of the Delannoy triangle. This method is also compared with known techniques of proving log-concavity.
Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser. A 85 (1999)] extended them to $q$-log-concavity and almost $q$-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomia…
We prove the log-concavity of the Fennessey-Larcombe-French sequence based on its three-term recurrence relation, which was recently conjectured by Zhao. The key ingredient of our approach is a sufficient condition for log-concavity of a sequence subject to certain three-term recurrence.
In this paper we develop a new geometric method to answer the log-concavity questions related to a nice class of combinatorial sequences arising from the Pascal triangle.
Ordinary polytopes are known as a non-simplicial generalization of the cyclic polytopes. The face vectors of ordinary polytopes are shown to be log-concave.
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.
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…
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).
We prove that in a large collection of naturally defined sets of permutations of fixed length, the numbers of permutations at Ulam distance k from the identity form a log-concave sequence in k.
This is a survey on recent developments on unimodality, log-concavity and real-rootedness in combinatorics. Stanley and Brenti have written extensive surveys of various techniques that can be used to prove real-rootedness, log-concavity or unimodality. After a brief introduction, we will complement these surveys with a survey over some new techniques that have been developed, as well as problems and conjectures that have been solved. This is a draft of a chapter to appear in Handbook of Enumerative Combinatorics, published by CRC Press.
This paper presents the log-concavity of the $m$-gonal figurate number sequences. The author gives and proves the recurrence formula for $m$-gonal figurate number sequences and its corresponding quotient sequences which are found to be bounded. Finally, the author also show that for $m\geq 3$, the sequence $\big \{S_n(m)\big\}_{n\geq 1}$ of $m$-gonal figurate numbers is a log-concave.
Let $\{P_n\}_{n\geq 0}$ denote the Catalan-Larcombe-French sequence, which naturally came up from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence $\{\sqrt[n]{P_n}\}_{n\geq 1}$, which was originally conjectured by Sun. We also obtain the strict log-concavity of the sequence $\{\sqrt[n]{V_n}\}_{n\geq 1}$, where $\{V_n\}_{n\geq 0}$ is the Fennessey-Larcombe-French sequence arising in the series expansion of the complete elliptic integral of the second kind.
We prove that the partition function $p(n)$ is log-concave for all $n>25$. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the Rademacher series for $p(n)$.
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…
Let $\{h_1,h_2,...\}$ be a set of algebraically independent variables. We ask which vectors are extreme in the cone generated by $h_ih_j-h_{i+1}h_{j-1}$ ($i\geq j>0$) and $h_i$ ($i>0$). We call this cone the cone of log concavity. More generally, we ask which vectors are extreme in the cone generated by Schur functions of partitions with $k$ or fewer parts. We give a conjecture for which vectors are extreme in the cone of log concavity. We prove the characterization in one direction and give partial results in the other direction.
Let the Bessel number of the second kind B(n,k) be the number of set partitions of [n] into k blocks of size one or two, and let the Bessel number of the first kind b(n,k) be a certain coefficient in n-th Bessel polynomial. In this paper, we show that Bessel numbers satisfy two properties of Stirling numbers: The two kinds of Bessel numbers are related by inverse formulas, and both Bessel numbers of the first kind and the second kind form log-concave sequences. By constructing sign-reversing involutions, we prove the inverse formulas. We review Krattenthaler's injection for the log-concavity o…
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…
D. Uminsky and K. Yeats [6] studied the properties of the log- operator L on the subset of the finite symmetric sequences and prove the existence of an infinite region R, bounded by parametrically de- fined hypersurfaces such that any sequence corresponding a point of R is infinitely log concave. We study the properties of a new operator L_r and redefine the hypersurfaces which generalizes the one defined by Uminsky and Yeats [6]. We show that any sequence corresponding a point of the region R, bounded by the new generalized parametrically defined r-factor hypersurfaces, is Generalized r-facto…
Let $e_{n}^k$ be the entries in the classical Euler's difference table. We consider the array $d_{n}^{k}=e_n^k/k!$ for $0\leq k \leq n$, where $d_n^k$ can be interpreted as the number of k-fixed-points-permutations of [n]. We show that the sequence $\{d_n^k\}_{0\leq k\leq n}$ is 2-log-concave and reverse ultra log-concave for any given n.
Given a sequence (a_k) = a_0, a_1, a_2,... of real numbers, define a new sequence L(a_k) = (b_k) where b_k = a_k^2 - a_{k-1} a_{k+1}. So (a_k) is log-concave if and only if (b_k) is a nonnegative sequence. Call (a_k) "infinitely log-concave" if L^i(a_k) is nonnegative for all i >= 1. Boros and Moll conjectured that the rows of Pascal's triangle are infinitely log-concave. Using a computer and a stronger version of log-concavity, we prove their conjecture for the nth row for all n <= 1450. We also use our methods to give a simple proof of a recent result of Uminsky and Yeats about regions of in…
In recent years, the log-concavity of $\{\sqrt[n]{S_n}\}_{n\geq 1}$ have been received a lot of attention. Very recently, Sun posed the following conjecture in his new book: the sequences $\{\sqrt[n]{a_n}\}_{n\geq 2}$ and $\{ \sqrt[n]{b_n}\}_{n\geq 1}$ are log-concave, where \[ a_n:= \frac{1}{n}\sum_{k=0}^{n-1} \frac{{n-1\choose k}^2{n+k\choose k}^2 }{4k^2-1} \] and \[ b_n:= \frac{1}{n^3}\sum_{k=0}^{n-1} (3k^2+3k+1){n-1\choose k}^2 {n+k\choose k}^2. \] In this paper, two methods, semi-automatic and analytic methods, are used to confirm Sun's conjecture. The semi-automatic method relies on a cr…
The chromatic polynomial is a well studied object in graph theory. There are many results and conjectures about the log-concavity of the chromatic polynomial and other polynomials related to it. The location of the roots of these polynomials has also been well studied. One famous result due to A. Sokal and C. Borgs provides a bound on the absolute value of the roots of the chromatic polynomial in terms of the highest degree of the graph. We use this result to prove a modification of a log-concavity conjecture due to F. Brenti. The original conjecture of Brenti was that the chromatic polynomial…
$q$-Kaplansky numbers were considered by Chen and Rota. We find that $q$-Kaplansky numbers are connected to the symmetric differences of Gaussian polynomials introduced by Reiner and Stanton. Based on the work of Reiner and Stanton, we establish the unimodality of $q$-Kaplansky numbers. We also show that $q$-Kaplansky numbers are the generating functions for the inversion number and the major index of two special kinds of $(0,1)$-sequences. Furthermore, we show that $q$-Kaplansky numbers are strongly $q$-log-concave.
We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called \textit{synchronicity} and \textit{ratio-dominance}, and a characterization of some bivariate sequences as \textit{lexicographic}. We are motivated by the 25-year old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distribut…
The classical Eulerian Numbers $A_{n,k}$ are known to be log-concave. Let $P_{n,k}$ and $Q_{n,k}$ be the number of even and odd permutations with $k$ excedances. In this paper, we show that $P_{n,k}$ and $Q_{n,k}$ are log-concave. For this, we introduce the notion of strong synchronisation and ratio-alternating which are motivated by the notion of synchronisation and ratio-dominance, introduced by Gross, Mansour, Tucker and Wang in 2014. We show similar results for Type B Coxeter Groups. We finish with some conjectures to emphasize the following: though strong synchronisation is stronger than …
Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave: 1. The matroid complex of independent subsets of E. 2. The broken circuit complex of $M$ relative to an ordering of E. The first implies a conjecture of Colbourn on the reliability polynomial of a graph, and the second implies a conjecture of Hoggar on the chromatic polynomial of a graph. The proof is based on the geometric formula for the characteristic polynomial of Denham, Garrousian, and Schulze.
In the $1970$s Nicolas proved that the coefficients $p_d(n)$ defined by the generating function \begin{equation*} \sum_{n=0}^{\infty} p_d(n) \, q^n = \prod_{n=1}^{\infty} \left( 1- q^n\right)^{-n^{d-1}} \end{equation*} are log-concave for $d=1$. Recently, Ono, Pujahari, and Rolen have extended the result to $d=2$. Note that $p_1(n)=p(n)$ is the partition function and $p_2(n)=\func{pp}\left( n\right) $ is the number of plane partitions. In this paper, we invest in properties for $p_d(n)$ for general $d$. Let $n \geq 6$. Then $p_d(n)$ is almost log-concave for $n$ divisible by $3$ and almost str…
Let $m$ and $k \geq 2$ be positive integers. We show that polynomial $P = (1+x)^m(1+x^k)$ is strongly unimodal (frequently known as {\it log concave\/}) if and only if $m \geq k^2 -3$; this is also the criterion for $P$ to be merely unimodal (that is, for $P$ of this form, unimodality implies strong unimodality).{ }In section 2, we investigate an analogous question, concerning the property $\EE$ of functions $f$ analytic on a neighbourhood of the unit circle [H2], and show that the corresponding minimal $m$ is rather surprisingly of order $k^4$.
We prove the reverse ultra log-concavity of the Boros-Moll polynomials. We further establish an inequality which implies the log-concavity of the sequence $\{i!d_i(m)\}$ for any $m\geq 2$, where $d_i(m)$ are the coefficients of the Boros-Moll polynomials $P_m(a)$. This inequality also leads to the fact that in the asymptotic sense, the Boros-Moll sequences are just on the borderline between ultra log-concavity and reverse ultra log-concavity. We propose two conjectures on the log-concavity and reverse ultra log-concavity of the sequence $\{d_{i-1}(m) d_{i+1}(m)/d_i(m)^2\}$ for $m\geq 2$.
We introduce the antipodal pairs property for probability measures on finite Boolean algebras and prove that conditional versions imply strong forms of log-concavity. We give several applications of this fact, including improvements of some results of Wagner; a new proof of a theorem of Liggett stating that ultra-log-concavity of sequences is preserved by convolutions; and some progress on a well-known log-concavity conjecture of J. Mason.
In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series. We propose several conjectures concerning zeros and coefficients of a generalized form of those polynomials. We also consider polynomials whose generating functions are higher order Toeplitz determinants formed by rising factorial series. We make three conjectures about these polynomials. All proposed conjectures are supported by numerical evidence.
We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which i…
Let $π$ be a permutation of $[n]=\{1,\dots,n\}$ and denote by $\ell(π)$ the length of a longest increasing subsequence of $π$. Let $\ell_{n,k}$ be the number of permutations $π$ of $[n]$ with $\ell(π)=k$. Chen conjectured that the sequence $\ell_{n,1},\ell_{n,2},\dots,\ell_{n,n}$ is log concave for every fixed positive integer $n$. We conjecture that the same is true if one is restricted to considering involutions and we show that these two conjectures are closely related. We also prove various analogues of these conjectures concerning permutations whose output tableaux under the Robinson-Sche…
A Ringel ladder can be formed by a self-bar-amalgamation operation on a symmetric ladder, that is, by joining the root vertices on its end-rungs. The present authors have previously derived criteria under which linear chains of copies of one or more graphs have log-concave genus polynomials. Herein we establish Ringel ladders as the first significant non-linear infinite family of graphs known to have log-concave genus polynomials. We construct an algebraic representation of self-bar-amalgamation as a matrix operation, to be applied to a vector representation of the partitioned genus distributi…
We expand upon the notion of equivariant log concavity, and make equivariant log concavity conjectures for Orlik--Solomon algebras of matroids, Cordovil algebras of oriented matroids, and Orlik--Terao algebras of hyperplane arrangements. In the case of the Coxeter arrangement for the Lie algebra $\mathfrak{sl}_n$, we exploit the theory of representation stability to give computer assisted proofs of these conjectures in low degree.
Let $\bar{p}(n)$ denote the overpartition function. In this paper, we study the asymptotic higher order $\log$-concavity property of the overpatition function in a similar framework done by Hou and Zhang for the partition function. This will enable us to move on further in order to prove $\log$-concavity of overpartitions, explicitly by studying the asymptotic expansion of the quotient $\bar{p}(n-1)\bar{p}(n+1)/\bar{p}(n)^2$ upto a certain order so that one can finally ends up with the phenomena of $2$-$\log$-concavity and higher order Turán property of $\bar{p}(n)$ by following a sort of unif…
