Related papers: On consecutive sums in permutations

We consider uniform random permutations in proper substitution-closed classes and study their limiting behavior in the sense of permutons. The limit depends on the generating series of the simple permutations in the class. Under a mild…

For a fixed positive integer $m$ and any partition $m = m_1 + m_2 + \cdots + m_e$ , there exists a sequence $\{n_{i}\}_{i=1}^{k}$ of positive integers such that $$m=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\cdots+\frac{1}{n_{k}},$$ with the property…

The following result gives the flavor of this paper: Let $t$, $k$ and $q$ be integers such that $q\geq 0$, $0\leq t < k$ and $t \equiv k \,({\rm mod}\, 2)$, and let $s\in [0,t+1]$ be the unique integer satisfying $s \equiv q +…

The average properties of the well-known Subset Sum Problem can be studied by the means of its randomised version, where we are given a target value $z$, random variables $X_1, \ldots, X_n$, and an error parameter $\varepsilon > 0$, and we…

Let $A_1, A_2, \ldots, A_n$ be events in a sample space. Given the probability of the intersection of each collection of up to $k+1$ of these events, what can we say about the probability that at least $r$ of the events occur? This question…

For integers $a_1, \dots, a_n \ge 0$ and $k \ge 1$, let $\mathcal L_{k+2}(a_1, \dots, a_n)$ denote the set of permutations of $\{1, \dots, a_1+\dots+a_n\}$ whose descent set is contained in $\{a_1, a_1+a_2, \dots, a_1+\dots+a_{n-1}\}$, and…

Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of…

We count the number of subsets of $\{1,2,\cdots,n\}$ under different conditions and study the sequence obtained as we let $n$ increase.

Let $T_o(k)$ denote the number of solutions of $\sum_{i=1}^k\frac 1{x_i}=1$ in odd numbers $1<x_1<x_2<...<x_k$. It is clear that $T_o(2k)=0$. For distinct primes $p_1, p_2,..., p_t$, let $S(p_1, p_2,...,…

We explore how the asymptotic structure of a random permutation of $[n]$ with $m$ inversions evolves, as $m$ increases, establishing thresholds for the appearance and disappearance of any classical, consecutive or vincular pattern. The…

We show that sequences of positive integers whose ratios $a_n^2/a_{n+1}$ lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only…

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…

We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random…

Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at…

We calculate the large deviations for the length of the longest alternating subsequence and for the length of the longest increasing subsequence in a uniformly random permutation that avoids a pattern of length three. We treat all six…

Let $m, n$ be positive integers such that $m>1$ divides $n$. In this paper, we introduce a special class of piecewise-affine permutations of the finite set $[1, n]:=\{1, \ldots, n\}$ with the property that the reduction $\pmod m$ of $m$…

Let $s$ denote West's stack-sorting map. A permutation is called $t-\textit{sorted}$ if it is of the form $s^t(\mu)$ for some permutation $\mu$. We prove that the maximum number of descents that a $t$-sorted permutation of length $n$ can…

We consider series of the form $$ \frac{p}{q} +\sum_{j=2}^\infty \frac{1}{x_j}, $$ where $x_1=q$ and the integer sequence $(x_n)$ satisfies a certain non-autonomous recurrence of second order, which entails that $x_n|x_{n+1}$ for $n\geq 1$.…