The Canonical Bijection Between Pipe Dreams and Bumpless Pipe Dreams

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   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.

2. 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.

3. Related Not relevant

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   Urban Larsson, Richard J. Nowakowski, Carlos P. Santos · 2024 · Combinatorics · AI summary available

   Abstract

   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…

4. Related Not relevant

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   Lutz Warnke · 2012 · Combinator. Probab. Comp. 25 (2016) 269-299 · AI summary available

   Abstract

   Concentration inequalities are fundamental tools in probabilistic combinatorics and theoretical computer science for proving that random functions are near their means. Of particular importance is the case where f(X) is a function of independent random variables X=(X_1, ..., X_n). Here the well known bounded differences inequality (also called McDiarmid's or Hoeffding-Azuma inequality) establishes sharp concentration if the function f does not depend too much on any of the variables. One attractive feature is that it relies on a very simple Lipschitz condition (L): it suffices to show that |f(X)-f(X')| \leq c_k whenever X,X' differ only in X_k. While this is easy to check, the main disadvant…

5. Related Not relevant

   Order Preserving Maps of Posets

   Zhousheng Mei, Suijie Wang · 2017 · Combinatorics · Not summarized yet

   Abstract

   For any two finite posets $P$ and $Q$, let $\Hom(P,Q)$ be the hom-poset consisting of all order preserving maps from $P$ to $Q$, and $J(Q)$ the collection of all order ideals of $Q$. In this paper, we study some basic properties of the hom-poset $\Hom(P,Q)$ and prove that $\Hom\big(P,J(Q)\big)$ is a distributive lattice and characterized by \[ \Hom\big(P,J(Q)\big)\cong J(P^*\times Q), \] where $P^*$ is the dual of $P$. Consequently, we obtain that $\Hom\big(P,J(Q)\big)$ and $\Hom\big(Q,J(P)\big)$ are dual isomorphic, i.e., \[ \Hom\big(P,J(Q)\big)\cong \Hom^{*}\big(Q,J(P)\big). \] As applications, we calculate the number of order preserving maps from any poset to the boolean algebra, and the …

6. Related Not relevant

   Isomorphisms between dense random graphs

   Erlang Surya, Lutz Warnke, Emily Zhu · 2023 · Combinatorics · AI summary available

   Abstract

   We consider two variants of the induced subgraph isomorphism problem for two independent binomial random graphs with constant edge-probabilities p_1,p_2. We resolve several open problems of Chatterjee and Diaconis, and also confirm simulation-based predictions of McCreesh, Prosser, Solnon and Trimble: (i) we prove a sharp threshold result for the appearance of G_{n,p_1} as an induced subgraph of G_{N,p_2}, (ii) we show two-point concentration of the maximum common induced subgraph of G_{N, p_1} and G_{N,p_2}, and (iii) we show that the number of induced copies of G_{n,p_1} in G_{N,p_2} has an unusual limiting distribution.

7. Related Not relevant

   The degree-restricted random process is far from uniform

   Michael Molloy, Erlang Surya, Lutz Warnke · 2022 · Combinatorics · AI summary available

   Abstract

   The degree-restricted random process is a natural algorithmic model for generating graphs with degree sequence D_n=(d_1, \ldots, d_n): starting with an empty n-vertex graph, it sequentially adds new random edges so that the degree of each vertex v_i remains at most d_i. Wormald conjectured in 1999 that, for d-regular degree sequences D_n, the final graph of this process is similar to a uniform random d-regular graph. In this paper we show that, for degree sequences D_n that are not nearly regular, the final graph of the degree-restricted random process differs substantially from a uniform random graph with degree sequence D_n. The combinatorial proof technique is our main conceptual contribu…

8. Related Not relevant

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

   Matthew Kwan, Lisa Sauermann · 2019 · Combinatorics · AI summary available

   Abstract

   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…