We obtain some new results on the unimodal sequences of the real values of rational functions by polynomials with positive integer coefficients. Thus, we introduce the notion of merged-log-concavity of rational functions. Roughly speaking, the notion extends Stanley's $q$-log-concavity of polynomials.
We construct explicit merged-log-concave rational functions by $q$-binomial coefficients, Hadamard products, and convolutions, extending the Cauchy-Binet formula. Then, we obtain the unimodal sequences of rational functions by Young diagrams. Moreover, we consider the variation of unimodal sequences by critical points that separate strictly increasing, strictly decreasing, and hill-shape sequences among almost strictly unimodal sequences. Also, the critical points are zeros of polynomials in a suitable setting.
The study above extends the $t$-power series of $(\pm t;q)_{\infty}^{\mp 1}$ to some extent by polynomials with positive integer coefficients and the variation of unimodal sequences. We then obtain the golden ratio of quantum dilogarithms ($q$-exponentials) as a critical point. Additionally, we consider eta products, generalized Narayana numbers, and weighted $q$-multinomial coefficients, which we introduce.
In statistical mechanics, we discuss the grand canonical partition functions of some ideal boson-fermion gases with or without Casimir energies (Ramanujan summation). The merged-log-concavity gives phase transitions on Helmholtz free energies by critical points of the metallic ratios including the golden ratio. In particular, the phase transitions implies non-zero particle vacua from zero particle vacua as the temperature rises.
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.
$q$-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.
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 will look at the Catalan numbers from the {\it Rigged Configurations} point of view originated \cite{Kir} from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our strategy is to take a combinatorial interpretation of Catalan numbers $C_n$ as the number of standard Young tableaux of rectangular shape $(n^2)$, or equivalently, as the Kostka number $K_{(n^2),1^{2n}}$, as the starting point of research. We observe that the rectangular (or multidimensional) Catalan numbers $ C(m,n)$ introduced and studied by P. MacMahon \cit…
In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems.
We study the $S_n$-equivariant log-concavity of the cohomology of flag varieties, also known as the coinvariant ring of $S_n$. Using the theory of representation stability, we give computer-assisted proofs of the equivariant log-concavity in low degrees and high degrees and conjecture that it holds for all degrees. Furthermore, we make a stronger unimodal conjecture which implies the equivariant log-concavity.
In this paper, we investigate two properties concerning the unimodality of the $δ$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\mathcal{P}$ of dimension $d$, we prove that the dilated lattice polytopes $n\mathcal{P}$ have strictly log-concave and strictly alternatingly increasing $δ$-vectors if $n > \max\{s,d+1-s\}$, where $s$ is the degree of the $δ$-polynomial of $\mathcal{P}$. The bound $\max\{s,d+1-s\}$ for $n$ is reasonable. We also provide several kinds of unimodal (or non-unimodal) $δ$-vectors. Concretely, we give exampl…
In this paper, we use the analytic method of Odlyzko and Richmond to study the log-concavity of power series. If $f(z) = \sum_n a_nz^n$ is an infinite series with $a_n \geq 1$ and $a_0 + \cdots + a_n = O(n + 1)$ for all $n$, we prove that a super-polynomially long initial segment of $f^k(z)$ is log-concave. Furthermore, if there exists constants $C > 1$ and $α< 1$ such that $a_0 + \cdots + a_n = C(n + 1) - R_n$ where $0 \leq R_n \leq O((n + 1)^α)$, we show that an exponentially long initial segment of $f^k(z)$ is log-concave. This resolves a conjecture proposed by Letong Hong and the author, w…
Unimodality of Independence Polynomials for Sufficiently Large Forests
Duke University（杜克大学）；Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）；Hebrew University（希伯来大学）；Universidad de Valladolid（巴利亚多利德大学）；University of Hawai‘i at Mānoa（夏威夷大学马诺阿分校）
In a previous paper, we studied an overpartition analogue of Gaussian polynomials as the generating function for overpartitions fitting inside an $m \times n$ rectangle. Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization $\overline{m+n \brack n}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of Delannoy numbers. First we obtain finite versions of classical $q$-series identities such as the $q$-binomial theorem and the Lebesgue identity, as well as two-variable generalizations of classical identities involving Gauss…
For every positive integer $n$, consider the linear operator $\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\frac{h(t)}{(1 - t)^{d + 1}} = \sum_{m \geq 0} g(m) t^{m}$, for some polynomial $g(m)$ with rational coefficients, then $\frac{\U_{n}h(t)}{(1- t)^{d + 1}} = \sum_{m \geq 0} g(nm) t^{m}$. We show that there exists a positive integer $n_{d}$, depending only on $d$, such that if $h(t)$ is a polynomial of degree at most $d$ with nonnegative integer coefficients and $h(0) \geq 1$, then for $n \geq n_{d}$, $\U_{n}h(t)$ has simple, real,…
The Links--Gould invariant $\mathrm{LG}(L ; t_0, t_1)$ of a link $L$ is a two-variable quantum generalization of the Alexander--Conway polynomial $Δ_L(t)$ and has been shown to share some of its most geometric features in several recent works. Here we suggest that $\mathrm{LG}$ likely shares another of the Alexander polynomial's most distinctive - and mysterious - properties: for alternating links, the coefficients of the Links-Gould polynomial alternate and appear to form a log-concave two-indexed sequence with no internal zeros. The former was observed by Ishii for knots with up to 10 crossi…
In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced by Guay-Paquet. We provide an algorithm which works for any function that satisfies this law, such as unicellular LLT polynomials. When the indifference graph has bipartite complement it reduces to a planar network, in this case, we prove that the coefficients of the chromatic quasisymmetric function in the elementary basis are positive unimodal polynomials and characterize them as certain $q$-hit numbers (up to a factor). Finally, we discuss the logarithmic conca…
We study Naruse-Newton coefficients, which are obtained from expanding descent polynomials in a Newton basis introduced by Jiradilok and McConville. These coefficients $C_0, C_1, \ldots$ form an integer sequence associated to each finite set of positive integers. For fixed nonnegative integers $a<b$, we examine the set $R_{a, b}$ of all ratios $\frac{C_a}{C_b}$ over finite sets of positive integers. We characterize finite sets for which $\frac{C_a}{C_b}$ is minimized and provide a construction to prove $R_{a, b}$ is unbounded above. We use this construction to obtain results on the closure of …
