A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.     

A unified framework to prove multiplicative inequalities for the partition function

In this paper, we consider a certain class of inequalities for the partition function of the following form:$\prod _{i=1}^{T}p(n+s_i)\ge \prod _{i=1}^{T}p(n+r_i),$ which we call multiplicative inequalities. Given a multiplicative inequality with the condition that ${\sum }_{i=1}^T{s}_i^m\ne {\sum }_{i=1}^T{r}_i^m$ for at least one $m\ge 1$, we shall construct a unified framework so as to decide whether such a inequality holds or not. As a consequence, we will see that study of such inequalities has manifold applications. For example, one can retrieve log-concavity property, strong log-concavity, and the multiplicative inequality for $p(n)$ considered by Bessenrodt and Ono, to name a few. Furthermore, we obtain an asymptotic expansion for the finite difference of the logarithm of $p(n)$, denoted by ${(-1)}^{r-1}{\mathrm{\Delta }}^r\log p(n)$, which generalizes a result by Chen, Wang, and Xie.     

On Serre dimension of monoid algebras and Segre extensions

Let R be a commutative noetherian ring of dimension d and M be a commutative, cancellative, torsion-free monoid of rank r. Then S-$dim(R[M])\le max\{1,dim(R[M])?1\}$. Further, we define a class of monoids ${\{{\mathfrak{M}}_n\}}_{n\ge 1}$ such that if $M\in {\mathfrak{M}}_n$ is seminormal, then S-$dim(R[M])\le dim(R[M])?n=d+r?n$, where $1\le n\le r$. As an application, we prove that for the Segre extension $S_{mn}(R)$ over R, S-$dim(S_{mn}(R))\le dim(S_{mn}(R))?\left[\frac{m+n?1}{min\{m,n\}}\right]=d+m+n?1?\left[\frac{m+n?1}{min\{m,n\}}\right]$.     

On discrete Brunn-Minkowski and isoperimetric type inequalities

We show that the lattice point enumerator ${\mathrm{G}}_n(?)$ satisfies${\mathrm{G}}_n{(tK+sL+{(?1,?t+s?)}^n)}^{1/n}\ge t{\mathrm{G}}_n{(K)}^{1/n}+s{\mathrm{G}}_n{(L)}^{1/n}$ for any $K,L?{\mathbb{R}}^n$ bounded sets with integer points and all $t,s\ge 0$.We also prove that a certain family of compact sets, extending that of cubes ${[?m,m]}^n$, with $m\in \mathbb{N}$, minimizes the functional ${\mathrm{G}}_n(K+t{[?1,1]}^n)$, for any $t\ge 0$, among those bounded sets $K?{\mathbb{R}}^n$ with given positive lattice point enumerator.Finally, we show that these new discrete inequalities imply the corresponding classical Brunn-Minkowski and isoperimetric inequalities for non-empty compact sets.     

Erdős matching conjecture for almost perfect matchings

In 1965 Erd?s asked, what is the largest size of a family of k-element subsets of an n-element set that does not contain a matching of size $s+1$? In this note, we improve upon a recent result of Frankl and resolve this problem for $s>101k^3$ and $(s+1)k?n<(s+1)(k+\frac{1}{100k})$.     

On the size of special class 1 graphs and (P3;k)-co-critical graphs

A well-known theorem of Vizing states that if G is a simple graph with maximum degree Δ, then the chromatic index ${\chi }^{\prime }(G)$ of G is Δ or $\mathrm{\Delta }+1$. A graph G is class 1 if ${\chi }^{\prime }(G)=\mathrm{\Delta }$, and class 2 if ${\chi }^{\prime }(G)=\mathrm{\Delta }+1$; G is Δ-critical if it is connected, class 2 and ${\chi }^{\prime }(G-e)<{\chi }^{\prime }(G)$ for every $e\in E(G)$. A long-standing conjecture of Vizing from 1968 states that every Δ-critical graph on n vertices has at least $(n(\mathrm{\Delta }-1)+3)/2$ edges. We initiate the study of determining the minimum number of edges of class 1 graphs G, in addition, ${\chi }^{\prime }(G+e)={\chi }^{\prime }(G)+1$ for every $e\in E(\bar{G})$. Such graphs have intimate relation to $(P_3;k)$-co-critical graphs, where a non-complete graph G is $(P_3;k)$-co-critical if there exists a k-coloring of $E(G)$ such that G does not contain a monochromatic copy of $P_3$ but every k-coloring of $E(G+e)$ contains a monochromatic copy of $P_3$ for every $e\in E(\bar{G})$. We use the bound on the size of the aforementioned class 1 graphs to study the minimum number of edges over all $(P_3;k)$-co-critical graphs. We prove that if G is a $(P_3;k)$-co-critical graph on $n\ge k+2$ vertices, then$e(G)\ge \frac{k}{2}(n-\lceil \frac{k}{2}\rceil -\epsilon )+\left(\begin{array}{c}\lceil k/2\rceil +\epsilon \\ 2\end{array}\right),$ where ε is the remainder of $n-\lceil k/2\rceil$ when divided by 2. This bound is best possible for all $k\ge 1$ and $n\ge \lceil 3k/2\rceil +2$.