Flips in symmetric separated set-systems

1999

We show that 2 log 2 (m) is the least number of symmetric differences that a family of m sets can produce. Furthermore we give two characterizations of the set-theoretic structure of the families for which that lower bound is actually attained.

2014

In this paper, generalizing those earlier results, we reveal wider classes of pure domains in $2^{[n]}$. This is obtained as a consequence of our study of a novel geometric--combinatorial model for weakly separated set-systems, so-called \emph{combined (polygonal) tilings} on a zonogon, which yields a new insight in the area.

International Mathematical Forum, 2013

We develop combinatorics of Fulton's essential set particularly with a connection to Baxter permutations. For this purpose, we introduce a new idea: dual essential sets. Together with the original essential set, we reinterpret Eriksson-Linusson's characterization of Baxter permutations in terms of colored diagrams on a square board. We also discuss a combinatorial structure on local moves of these essential sets under weak order on the symmetric groups. As an application, we extend several familiar results on Bruhat order for permutations to alternating sign matrices: We establish an improved criterion of Bruhat-Ehresmann order as well as Generalized Lifting Property using bigrassmannian permutations, a certain subclass of Baxter permutations.

Graphs and Combinatorics, 2014

Union-free families of subsets of [n] = {1, . . . n} have been studied in [2]. In this paper, we provide a complete characterization of maximal symmetric difference-free families of subsets of [n]. |A e,o | = 2 n−2 − x := u; |A e,e | = x |A o,e | = 2 n−2 − y := v; |A o,o | = y,

The Electronic Journal of Combinatorics

We call a quadruple $\mathcal{W}:=\langle F,U,\Omega,\Lambda \rangle$, where $U$ and $\Omega$ are two given non-empty finite sets, $\Lambda$ is a non-empty set and $F$ is a map having domain $U\times \Omega$ and codomain $\Lambda$, a pairing on $\Omega$. With this structure we associate a set operator $M_{\mathcal{W}}$ by means of which it is possible to define a preorder $\ge_{\mathcal{W}}$ on the power set $\mathcal{P}(\Omega)$ preserving set-theoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice $\mathbb{L}$ there exist a finite set $\Omega_{\mathbb{L}}$ and a pairing $\mathcal{W}$ on $\Omega_\mathbb{L}$ such that the quotient of the preordered set $(\mathcal{P}(\Omega_\mathbb{L}), \ge_\mathcal{W})$ with respect to its symmetrization is a lattice that is order-isomorphic to $\mathbb{L}$. In the second result, we prove that when the lattice $\mathbb{L}$ is endowed with an order-reversing involutory...

Discrete Mathematics, 2000

n are subsets of the symmetric and the hyperoctahedral groups, consisting of permutations which avoid certain patterns. The order relation is given by (strict) containment of the descent sets. In each case, by means of an explicit order-preserving bijection, we show that the poset of restricted permutations is an extension of the refinement order on noncrossing partitions. Several structural properties

Order, 2002

The Electronic Journal of Combinatorics

Given a space $\Omega$ endowed with symmetry, we define $ms(\Omega, r)$ to be the maximum of $m$ such that for any $r$-coloring of $\Omega$ there exists a monochromatic symmetric set of size at least $m$. We consider a wide range of spaces $\Omega$ including the discrete and continuous segments $\{1, \ldots, n\}$ and $[0,1]$ with central symmetry, geometric figures with the usual symmetries of Euclidean space, and Abelian groups with a natural notion of central symmetry. We observe that $ms(\{1, \ldots, n\}, r)$ and $ms([0,1], r)$ are closely related, prove lower and upper bounds for $ms([0,1], 2)$, and find asymptotics of $ms([0,1], r)$ for $r$ increasing. The exact value of $ms(\Omega, r)$ is determined for figures of revolution, regular polygons, and multi-dimensional parallelopipeds. We also discuss problems of a slightly different flavor and, in particular, prove that the minimal $r$ such that there exists an $r$-coloring of the $k$-dimensional integer grid without infinite mon...

Journal of Combinatorial Theory, Series A, 1989

We define a new poset on the symmetric group S,. It is a subposet of the weak ordering of S, with the property that every interval is a distributive lattice. For each principal ideal we explicitly compute the poset of join-irreducibles and use this to get an expression for the number of maximal chains in these intervals. In the case of certain principal ideals the number of maximal chains is given by the number of shifted tableaux of a certain shape. In particular, in the principal ideal generated by the reverse permutation the number of maximal chains is given by the number of shifted tableaux of staircase shape. We relate this to similar results for the weak ordering and show how bijections that work in that case restrict to this new poset. This new poset arises naturally from the study of convex geometries. We detail these connections and give an alternative definition for this poset in this context.

2012

In this dissertation we first study partition posets and their topology. For each composition c we show that the order complex of the poset of pointed set partitions is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition c. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module of a border strip associated to the composition. We also study the filter of pointed set partitions generated by knapsack integer partitions. In the second half of this dissertation we study descent avoidance in permutations. We extend the notion of consecutive pattern avoidance to considering sums over all permutations where each term is a product of weights depending on each consecutive pattern of a fixed length. We study the problem of finding the asymptotics of these sums. Our technique is to extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends on the descent...