Related papers: Ascent sequences and 3-nonnesting set partitions

In 1996, my brilliant student John Noonan, discovered, and proved that there are 3(2n)!/(n(n+3)!(n-3)!) ways to line-up n people of different heights in such a way that out of the n(n-1)(n-2)/6 possible triples of people exactly one is such…

A set partition avoids a pattern if no subdivision of that partition standardizes to the pattern. There exists a bijection between set partitions and restricted growth functions (RGFs) on which Wachs and White defined four statistics of…

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two…

For real numbers a, b> 1, let as a_b denote the result of interpreting a in base b instead of base 10. We define ``dungeons'' (as opposed to ``towers'') to be numbers of the form a_b_c_d_..._e, parenthesized either from the bottom upwards…

In this note we are interested in the problem of whether or not every increasing sequence of positive integers $x_1x_2x_3...$ with bounded gaps must contain a double 3-term arithmetic progression, i.e., three terms $x_i$, $x_j$, and $x_k$…

We study the integer sequence v_n of numbers of lines in hypersurfaces of degree 2n-3 of P^n, n>1. We prove a number of congruence properties of these numbers of several different types. Furthermore, the asymptotics of the v_n are described…

Given a sequence $S=(s_1,\dots,s_m) \in [0, 1]^m$, a block $B$ of $S$ is a subsequence $B=(s_i,s_{i+1},\dots,s_j)$. The size $b$ of a block $B$ is the sum of its elements. It is proved in [1] that for each positive integer $n$, there is a…

Permutations whose prefixes contain at least as many ascents as descents are called ballot permutations. Lin, Wang, and Zhao have previously enumerated ballot permutations avoiding small patterns and have proposed the problem of enumerating…

It is a classical result in combinatorics that among lattice paths with 2m steps U=(1,1) and D=(1,-1) starting at the origin, the number of those that do not go below the x-axis equals the number of those that end on the x-axis. A much more…

We consider a sorting machine consisting of two stacks in series where the first stack has the added restriction that entries in the stack must be in decreasing order from top to bottom. The class of permutations sortable by this machine…

A non-crossing spanning tree of a set of points in the plane is a spanning tree whose edges pairwise do not cross. Avis and Fukuda in 1996 proved that there always exists a flip sequence of length at most $2n-4$ between any pair of…

We improve the lower bound on the number of permutations of {1,2,...,n} in which no 3-term arithmetic progression occurs as a subsequence, and derive lower bounds on the upper and lower densities of subsets of the positive integers that can…

In their thought-provoking paper [1], Belkin et al. illustrate and discuss the shape of risk curves in the context of modern high-complexity learners. Given a fixed training sample size $n$, such curves show the risk of a learner as a…

We present four constructions of inversion sequences, and use them to compute the enumeration sequences of 24 classes of pattern-avoiding inversion sequences. This completes the enumeration of inversion sequences avoiding one or two…

In this paper we show a a proof by explicit bijections of the famous Kirkman-Cayley formula for the number of dissections of a convex polygon. Our starting point is the bijective correspondence between the set of nested sets made by \(k\)…

In 1930s Paul Erdos conjectured that for any positive integer C in any infinite +1 -1 sequence (x_n) there exists a subsequence x_d, x_{2d}, ... , x_{kd} for some positive integers k and d, such that |x_d + x_{2d} + ... + x_{kd}|> C. The…

Inversion sequences are integer sequences $(\sigma_1, \dots, \sigma_n)$ such that $0 \leqslant \sigma_i < i$ for all $1 \leqslant i \leqslant n$. The study of pattern-avoiding inversion sequences began in two independent articles by…

We introduce a new class of problems lying halfway between questions about graph capacity and intersection. We say that two binary sequences x and y of the same length have a skewincidence if there is a coordinate i for which x_i=y_{i+1}=1…

A sequence $(x_n)_{n=1}^{\infty}$ on the torus $\mathbb{T}$ exhibits Poissonian pair correlation if for all $s\geq0$, \begin{equation*} \lim_{N\to\infty} \frac{1}{N}\#\left\{1\leq m\neq n \leq N : |x_m-x_n| \leq \frac{s}{N}\right\} = 2s.…

Let us call a sequence of numbers heapable if they can be sequentially inserted to form a binary tree with the heap property, where each insertion subsequent to the first occurs at a leaf of the tree, i.e. below a previously placed number.…