We will try to sketch Professor F. Y. Wu's contributions in lattice statistical mechanics, solid state physics, graph theory, enumerative com-binatorics and so many other domains of physics and mathematics. We will recall F. Y. Wu's most important and well-known classic results and we will also sketch his most recent researches dedicated to the connections of lattice statistical mechanical mode...

In this paper we explore the combinatorics of the nonnegative part (G/P )≥0 of a cominuscule Grassmannian. For each such Grassmannian we define Γ -diagrams — certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P )≥0. In the classical cases, we describe Γ -diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can...

Abstract We present a proof of a conjecture about the relationship between Baxter permutations and pairs of alternating sign matrices that are produced from domino tilings of Aztec diamonds. It is shown that a tiling corresponds to a pair of ASMs that are both permutation matrices if and only if the larger permutation matrix corresponds to a Baxter permutation. There has been a thriving literat...

We propose a theoretical framework for a network covert channel based on enumerative combinatorics. It offers protocol independence and avoids detection by using a mimicry defense. Using a network monitoring phase, traffic is analyzed to detect which application-layer protocols are allowed through the firewalls. Using these results, a covert channel is built based on permutations of benign netw...

In this paper, we build on recent results by Chauve et al. and Bahrani and Lumbroso, which combined the splitdecomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphs—in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic). Our goal was to study a simple family of graphs, of...

Many combinatorial problems can be framed as counting the number of ways to allocate balls to urns, subject to various conditions. Richard Stanley invented the \twelvefold way" to organize these results into a table with twelve entries. See his book Enumerative Combinatorics, Volume 1. Let b represent the number of balls available and u the number of urns. The following table gives the number o...

A seminal technique of theoretical physics calledWick’s theorem interprets the Gaussian matrix integral of the products of the trace of powers of Hermitian matrices as the number of labelled maps with a given degree sequence, sorted by their Euler characteristics. This leads to the map enumeration results analogous to those obtained by combinatorial methods. In this paper we show that the enume...

1. Introduction. It is a lovely fact that n] = f1; 2; : : : ; ng, n 1, has as many subsets X with an even cardinality jXj as those with an odd cardinality, namely 2 n?1 of both. To prove it, pair every subset X with X 1 where X 1 is Xnf1g if 1 2 X and X f1g if 1 6 2 X. Then X 7 ! X 1 is an involution that changes the parity of jXj and the result follows. More generally, in enumerative combinato...

Motivated by higher vanishing multiplicity generalizations of Alon's Combinatorial Nullstellensatz and its applications, we study the following problem: for fixed k ? 1 $k\geqslant 1$ n $n$ large with respect to $k$ , what is minimum possible degree a polynomial P ? R [ x ? ] $P\in \mathbb {R}[x_1,\dots ,x_n]$ ( 0 ) ? $P(0,\dots ,0)\ne 0$ such that $P$ has zeroes at least all points in { } ? $\...

This article, based on joint work with Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, Dylan Thurston, and Rui Viana, presents a combinatorial model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky’s cluster algebras of...