Filling a box with translates of two bricks

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1. Related Not relevant

   Quasirandom Latin squares

   Jacob W. Cooper, Daniel Kral, Ander Lamaison et al. · 2020 · Combinatorics · Not summarized yet

   Abstract

   We prove a conjecture by Garbe et al. [arXiv:2010.07854] by showing that a Latin square is quasirandom if and only if the density of every 2x3 pattern is 1/720+o(1). This result is the best possible in the sense that 2x3 cannot be replaced with 2x2 or 1xN for any N.

2. Related Not relevant

   Enclosing Depth and other Depth Measures

   Patrick Schnider · 2021 · Combinatorics · Not summarized yet

   Abstract

   We study families of depth measures defined by natural sets of axioms. We show that any such depth measure is a constant factor approximation of Tukey depth. We further investigate the dimensions of depth regions, showing that the Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth measures which satisfy our most restrictive set of axioms, which includes Tukey depth. Along the way, we introduce and study a new depth measure called enclosing depth, which we believe to be of independent interest, and show its relation to a constant-fraction Radon theorem on certain two-colored point sets.

3. Related Not relevant

   A Generalization of a Theorem of Mandel

   Hery Randriamaro · 2023 · Combinatorics · AI summary available

   Abstract

   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.

4. Related Not relevant

   Almost generalized uniform matroids and excluded minors

   Hyungju Park · 2023 · Combinatorics · AI summary available

   Abstract

   We establish that matroids characterized by the Tutte polynomial $\sum_{i,j\ge 0}t_{i,j}x^iy^j$ with coefficients $t_{i,j}$ vanishing for $(i,j)\ge (k,l)$ precisely coincide with $(k,l)$-uniform matroids. This characterization implies that almost $(k,l)$-uniform matroids are exactly matroids with $t_{k,l}\le 1$ and $t_{i,j}=0$ if $(i,j)>(k,l)$. We also characterize excluded minors of almost $(k,l)$-uniform matroids in terms of Tutte polynomial coefficients. Finally, we construct an infinite family of excluded minors of almost $(k,l)$-uniform matroids which extend previously known cases of almost uniform and almost paving matroids.

5. Related Not relevant

   Symbolic Sets for Proving Bounds on Rado Numbers

   Tanbir Ahmed, Lamina Zaman, Curtis Bright · 2025 · Combinatorics · AI summary available

   Abstract

   Given a linear equation $\cal E$ of the form $ax + by = cz$ where $a$, $b$, $c$ are positive integers, the $k$-colour Rado number $R_k({\cal E})$ is the smallest positive integer $n$, if it exists, such that every $k$-colouring of the positive integers $\{1, 2, \dotsc, n\}$ contains a monochromatic solution to $\cal E$. In this paper, we consider $k = 3$ and the linear equations $ax + by = bz$ and $ax + ay = bz$. Using SAT solvers, we compute a number of previously unknown Rado numbers corresponding to these equations. We prove new general bounds on Rado numbers inspired by the satisfying assignments discovered by the SAT solver. Our proofs require extensive case-based analyses that are diff…

6. Related Not relevant

   Canonical Forms of $2 \times 2 \times 2$ and $2 \times 2 \times 2 \times 2$ Symmetric Tensors over Prime Fields

   Stavros G. Stavrou · 2012 · Combinatorics · AI summary available

   Abstract

   We consider symmetric tensors of format $2 \times 2 \times 2$ and $2 \times 2 \times 2 \times 2$ over prime fields. Using computer algebra we compute the canonical forms of these tensors. For $2 \times 2 \times 2$ symmetric tensors, we consider the prime fields $\mathbb{F}_p$ for $p = 2, 3, 5, 7, 11, 13, 17$. For $2 \times 2 \times 2 \times 2$ symmetric tensors, we consider the prime fields $\mathbb{F}_p$ for $p = 2, 3, 5, 7$. For each canonical form, we determine the size of its orbit and the symmetric rank of the symmetric tensors in its orbit.

7. Related Not relevant

   Lagrange Inversion Formula by Induction

   Erlang Surya, Lutz Warnke · 2023 · The American Mathematical Monthly 130 (2023), 944-948 · AI summary available

   Abstract

   We present a simple inductive proof of the Lagrange Inversion Formula.

8. Related Not relevant

   In How Many Ways Can You Reassemble Several Russian Dolls?

   Doron Zeilberger · 2009 · Combinatorics · Not summarized yet

   Abstract

   When my brilliant student Thotsaporn "Aek" Thanatipanonda asked me about how many ways can one cover n identical twins, it rang a Bell back to 1981 (see the article). As usual, the method of teaching the computer how to do its own combinatorics, and to use "analysis" (that is really part of discrete math) to enumerate challenging combinatorial objects is even more interesting than the theorems and even the algorithm. Of course, formulas are dead, long live algorithms! But even algorithms are dead, long live meta-algorithms!!