Narrow Results

Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ with even (resp. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound, we show that $M_k(n-1)+M_k(n+1)>2M_k(n)$ for $k=0$ or $1$ and $n\ge 39$. This result can be seen as the refinement of the classical result regarding the convexity of the partition function $p(n)$, which counts the number of partitions of $n$. We also show that $M_0(n)$ (resp. $M_1(n)$) is log-concave for $n\ge 94$ and satisfies the higher order Turán inequalities for $n\ge 207$ with the aid of the upper bound and the lower bound for $M_0(n)$ and $M_1(n)$.

The degenerate Bernoulli numbers β_n(λ) are polynomials with rational coefficients of degree n in the variable λ, which arise in several combinatorial settings. An appropriate change of variable transforms β_n(λ) into a polynomial whose coefficients are all positive. Here, we prove that this transformed polynomial is log-concave, and therefore unimodal. As a consequence, we deduce bounds on the absolute values of the roots of β_n(λ).

In this paper, we discuss the log-behavior of the Catalan–Larcombe–French sequence {P_n}_n≥0. We prove that {P_n}_n≥0 is log-balanced and {P_n/(n!)²}_n≥0 is unimodal. In addition, we show that {P_k/k!}_0≤k≤n is reverse ultra log-concave.

We establish discrete and continuous log-concavity results for a biparametric extension of the $q$-numbers and of the $q$-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Turán’s inequality.

In this paper, we examine the number of ways to partition an integer $n$ into $k$th powers when $n$ is large. Simplified proofs of some asymptotic results of Wright are given using the saddle-point method, including exact formulas for the expansion coefficients. The convexity and log-concavity of these partitions is shown for large $n$, and the stronger conjectures of Ulas are proved. The asymptotics of Wright’s generalized Bessel functions are also treated.

In this paper, we expand on the remark by Andrews on the importance of infinite sums and products in combinatorics. Let ${\{g_d(n)\}}_{d\ge 0,n\ge 1}$ be the double sequences ${\sigma }_d(n)={\sum }_{\ell |n}{\ell }^d$ or ${\psi }_d(n)=n^d$. We associate double sequences $\{p^{g_d}(n)\}$ and $\{q^{g_d}(n)\}$, defined as the coefficients of

Bessenrodt and Ono, Chen, Wang and Jia, DeSalvo and Pak were the first to discover the log-subadditivity, the log-concavity and the third-order Turán inequality of the partition function $p(n)$, respectively. Many other important partition statistics have been proved to enjoy similar properties. This paper focuses on the partition function ${\overline{p}}_k(n)$, which counts the number of overpartitions of n with no parts divisible by k. We provide a combinatorial proof to establish that for any $k\ge 3$, the partition function ${\overline{p}}_k(n)$ exhibits the strict log-subadditivity. Specifically, we show that ${\overline{p}}_k(a){\overline{p}}_k(b)>{\overline{p}}_k(a+b)$ for integers $a\ge b\ge 1$ and $a+b\ge k$. Furthermore, we investigate the log-concavity and the third-order Turán inequality for ${\overline{p}}_k(n)$, where $2\le k\le 9$.

We study the log-concavity of a sequence of $p,q$-binomial coefficients located on a ray of the $p,q$-Pascal triangle for certain directions, and we establish the preserving log-concavity of linear transformations associated to $p,q$-Pascal triangle.

Let $G$ be a graph of order $n$. A subset $S$ of $V(G)$ is a dominating set of $G$ if every vertex in $V(G)\setminus S$ is adjacent to at least one vertex of $S$. The domination polynomial of $G$ is the polynomial $D(G,x)={\sum }_{i=\gamma (G)}^{n}d(G,i)x^i$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$, and $\gamma (G)$ is the size of a smallest dominating set of $G$, called the domination number of $G$. Motivated by a conjecture in [S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, ARS Combin. 114 (2014) 257–266] which states that the domination polynomial of any graph is unimodal, we obtain sufficient conditions for this conjecture to hold. Also we study the unimodality of graph $G$ with $d(G,\gamma (G))=\gamma (G)+k$, where $k$ is an integer.

We establish discrete and continuous log-concavity results for a biparametric extension of the q-numbers and of the q-binomial coefficients. By using classical results for the Jacobi theta function we are able to lift some of our log-concavity results to the elliptic setting. One of our main ingredients is a putatively new lemma involving a multiplicative analogue of Turán’s inequality.