Let ${\mathbb{F}}_2^n$ be the finite field of cardinality $2^n$. For all large $n$, any subset $A\subset {\mathbb{F}}_2^n\times {\mathbb{F}}_2^n$ of cardinality
$$|A|\gtrsim \frac{4^n\log \log n}{\log n},$$
must contain three points $\{(x,y),(x+d,y),(x,y+d)\}$ for $x,y,d\in {\mathbb{F}}_2^n$ and $d\ne 0$. Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on $\log n$, which is larger than has been obtained previously.

In 1972, Bender and Knuth established a bijection between certain infinite matrices of non-negative integers and plane partitions and in [2] a bijection between Bender-Knuth matrices and n-color partitions was shown. Here we use this later bijection and translate the recently found n-color partition theoretic interpretations of four mock theta functions of S. Ramanujan in [1] to new combinatorial interpretations of the same mock theta functions involving Bender-Knuth matrices.

We present analytical properties of a sequence of integers related to the evaluation of a rational integral. We also discuss an algorithm for the evaluation of the 2-adic valuation of these integers that has a combinatorial interpretation.

It is proposed that finding the recursion relation and generating function for the (colored) Motzkin numbers of higher rank introduced recently is an interesting problem.

Let ${\mathbb{F}}_2^n$ be the finite field of cardinality $2^n$. For all large $n$, any subset $A\subset {\mathbb{F}}_2^n\times {\mathbb{F}}_2^n$ of cardinality $$|A|\gtrsim \frac{4^n\log \log n}{\log n},$$ must contain three points $\{(x,y),(x+d,y),(x,y+d)\}$ for $x,y,d\in {\mathbb{F}}_2^n$ and $d\ne 0$. Our argument is an elaboration of an argument of Shkredov [14], building upon the finite field analog of Ben Green [10]. The interest in our result is in the exponent on $\log n$, which is larger than has been obtained previously.

Let $S$ be a finite set of positive integers with largest element $m$. Let us randomly select a composition $a$ of the integer $n$ with parts in $S$, and let $m(a)$ be the multiplicity of $m$ as a part of $a$. Let $0\le r<q$ be integers, with $q\ge 2$, and let $p_{n,r}$ be the probability that $m(a)$ is congruent to $r\mathrm{mod}q$. We show that if $S$ satisfies a certain simple condition, then $\underset{n\to \mathrm{\infty }}{lim}{p}_{n,r}=1/q$. In fact, we show that an obvious necessary condition on $S$ turns out to be sufficient.