Lagrange Inversion Formula by Induction

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   An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

   Matthew Kwan, Lisa Sauermann · 2019 · Combinatoire · Résumé IA disponible

   Résumé scientifique

   Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger…

2. Apparenté Pas pertinent

   Filling a box with translates of two bricks

   Mihail N. Kolountzakis · 2004 · Combinatoire · Résumé IA disponible

   Résumé scientifique

   We give a new proof of the following interesting fact recently proved by Bower and Michael: if a d-dimensional rectangular box can be tiled using translates of two types of rectangular bricks, then it can also be tiled in the following way. We can cut the box across one of its sides into two boxes, one of which can be tiled with the first brick only and the other one with the second brick. Our proof relies on the Fourier Transform. We also show that no such result is true for three, or more, types of bricks.

3. Apparenté Pas pertinent

   Using a Grassmann graph to recover the underlying projective geometry

   Ian Seong · 2023 · Graphs and Combinatorics 40, 88 (2024) · Résumé IA disponible

   Résumé scientifique

   Let $n,k$ denote integers with $n>2k\geq 6$. Let $\mathbb{F}_q$ denote a finite field with $q$ elements, and let $V$ denote a vector space over $\mathbb{F}_q$ that has dimension $n$. The projective geometry $P_q(n)$ is the partially ordered set consisting of the subspaces of $V$; the partial order is given by inclusion. For the Grassman graph $J_q(n,k)$ the vertex set consists of the $k$-dimensional subspaces of $V$. Two vertices of $J_q(n,k)$ are adjacent whenever their intersection has dimension $k-1$. The graph $J_q(n,k)$ is known to be distance-regular. Let $\partial$ denote the path-length distance function of $J_q(n,k)$. Pick two vertices $x,y$ in $J_q(n,k)$ such that $1<\partial(x,y)<…

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   The Canonical Bijection Between Pipe Dreams and Bumpless Pipe Dreams

   Yibo Gao, Daoji Huang · 2021 · Combinatoire · Résumé IA disponible

   Résumé scientifique

   We present a direct bijection between reduced pipe dreams and reduced bumpless pipe dreams by interpreting reduced compatible sequences on bumpless pipe dreams and show that this bijection preserves Monk's formula, establishing its canonical nature.

5. Apparenté Pas pertinent

   Affine Normal Play

   Urban Larsson, Richard J. Nowakowski, Carlos P. Santos · 2024 · Combinatoire · Résumé IA disponible

   Résumé scientifique

   There are many combinatorial games in which a move can terminate the game, such as a checkmate in chess. These moves give rise to diverse situations that fall outside the scope of the classical normal play structure. To analyze these games, an algebraic extension is necessary, including infinities as elements. In this work, affine normal play, the algebraic structure resulting from that extension, is analyzed. We prove that it is possible to compare two affine games using only their forms. Furthermore, affine games can still be reduced, although the reduced forms are not unique. We establish that the classical normal play is order-embedded in the extended structure, constituting its substruc…

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   A Generalization of a Theorem of Mandel

   Hery Randriamaro · 2023 · Combinatoire · Résumé IA disponible

   Résumé scientifique

   A theorem of Mandel allows to determine the covector set of an oriented matroid from its set of topes by using the composition condition. We provide a generalization of that result, stating that the covector set of a conditional oriented matroid can also be determined by its set of topes, but by using the face symmetry condition. It permits to represent geometrical configurations in terms of conditional oriented matroids, more suitable for computer calculations. We treat apartments of hyperplane arrangements as example.

7. Apparenté Pas pertinent

   Proof of the Kakeya set conjecture over rings of integers modulo square-free $N$

   Manik Dhar, Zeev Dvir · 2020 · Combinatoire · Pas encore résumé

   Résumé scientifique

   A Kakeya set $S \subset (\mathbb{Z}/N\mathbb{Z})^n$ is a set containing a line in each direction. We show that, when $N$ is any square-free integer, the size of the smallest Kakeya set in $(\mathbb{Z}/N\mathbb{Z})^n$ is at least $C_{n,\epsilon} N^{n - \epsilon}$ for any $\epsilon$ -- resolving a special case of a conjecture of Hickman and Wright. Previously, such bounds were only known for the case of prime $N$. We also show that the case of general $N$ can be reduced to lower bounding the $\mathbb{F}_p$ rank of the incidence matrix of points and hyperplanes over $(\mathbb{Z}/p^k\mathbb{Z})^n$.

8. Apparenté Pas pertinent

   De Moivre and Bell polynomials

   Cormac O'Sullivan · 2022 · Combinatoire · Résumé IA disponible

   Résumé scientifique

   We survey a family of polynomials that are very useful in all kinds of power series manipulations, and appearing more frequently in the literature. Applications to formal power series, generating functions and asymptotic expansions are described, and we discuss the related work of De Moivre, Arbogast and Bell.