Related papers: On a conjecture about enumerating $(2+2)$-free pos…

In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n.…

We introduce a new family of finite posets which we call 2-chains. These first arose in the study of 0-Hecke algebras, but they admit a variety of different characterisations. We give these characterisations, prove that they are equivalent…

Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced…

Let $I_n(x)=\prod_{i=1}^n \left( 1+x^{F_{i+1}}\right)$, where $F_{i+1}$ denotes a Fibonacci number. Let $v_r(n)$ denote the sum of the $r$th powers of the coefficients of $I_n(x)$. Our prototypical result is that $\sum_{n\geq 0} v_2(n)x^n=…

We show that there is a constant $C>0$ such that for each integer $n\geq 1$, there is a poset on at most $2^{2n/3+C\sqrt{n}}$ elements that contains each $n$-element poset as an (induced) subposet.

The aim of the present paper is to show that the concept of intuitionistic logic based on a Heyting algebra can be generalized in such a way that it is formalized by means of a bounded poset. In this case it is not assumed that the poset is…

We propose a general conjecture on decompositions of finite simple groups as products of conjugates of an arbitrary subset. We prove this conjecture for bounded subsets of arbitrary finite simple groups, and for large subsets of groups of…

In 1961, P. Erd\H{o}s, A. Ginzburg, and A. Ziv proved a remarkable theorem stating that each set of $2n-1$ integers contains a subset of size $n$, the sum of whose elements is divisible by $n$. We will prove a similar result for pairs of…

We study combinatorial inequalities for various classes of set systems: matroids, polymatroids, poset antimatroids, and interval greedoids. We prove log-concavity inequalities for counting certain weighted feasible words, which generalize…

In this work we are interested in the Demyanov--Ryabova conjecture for a finite family of polytopes. The conjecture asserts that after a finite number of iterations (successive dualizations), either a 1-cycle or a 2-cycle eventually comes…

Amdeberhan conjectured that the number of $(t,t+1, t+2)$-core partitions is $\sum_{0\leq k\leq [\frac{t}{2}]}\frac{1}{k+1}\binom{t}{2k}\binom{2k}{k}$. In this paper, we obtain the generating function of the numbers $f_t$ of $(t, t + 1, ...,…

For a finite set of non-zero natural numbers that contains at least one element different from 1 and the least common multiple of any of its subsets, there exists a subset of at least half of its members which has a common divisor larger…

This paper exploits adjacencies between the orbits of an ordered set P and a consequence of the classification of finite simple groups to, in many cases, exponentially bound the number of automorphisms. Results clearly identify the…

n this paper, we state several conjectures regarding distribution of primes and of pairs of primes represented by irreducible homogeneous polynomial in two variables $f(a,b)$. We formulate conjectures with respect to the slope $t=b/a$ for…

The Scholz conjecture on addition chains states that $\ell(2^n-1) \leq \ell(n) + n -1$ for all integers $n$ where $\ell(n)$ stands for the minimal length of all addition chains for $n$. It is proven to hold for infinite sets of integers. In…

We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements,…

Given a complemented poset P, we can assign to every element x of P the set x^+ of all its complements. We study properties of the operator ^+ on P, in particular, we are interested in the case when x^+ forms an antichain or when ^+ is…

In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the…

In this article we show that any free hyperplane arrangement with exponents 1's and 2's is a supersolvable arrangement. We conjecture that any free arrangement with exponents 1's, 2's and exactly one 3, is also supersolvable, and we show…