scholarly journals Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

On Lattice Embeddings for Partially Ordered Sets

1954 ◽  
Vol 6 ◽  
pp. 525-528
Author(s):  
Truman Botts
Keyword(s):  
Binary Relation ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
The Family ◽  
Transitive Binary Relation

Let P be a set partially ordered by a (reflexive, antisymmetric, and transitive) binary relation ≺. Let be the family of all subsets K of P having the property that x ∈ P and y ∈ K and y ≺ x imply x ∈ K.

scholarly journals Edgewise Cohen-Macaulay connectivity of partially ordered sets

2018 ◽  
Vol 122 (1) ◽  
pp. 5
Author(s):  
Christos A. Athanasiadis ◽  
Myrto Kallipoliti
Keyword(s):  
Closed Interval ◽  
Convex Polytopes ◽  
Partially Ordered Sets ◽  
Strong Form ◽  
Hasse Diagram ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Open Questions

The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen-Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen-Macaulay poset of the same rank. A corresponding notion of edgewise Cohen-Macaulay connectivity for partially ordered sets is investigated. Examples and open questions are discussed.

scholarly journals Note on a Ramsey Theorem for Posets with Linear Extensions

10.37236/6392 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Andrii Arman ◽  
Vojtěch Rödl
Keyword(s):  
Short Proof ◽  
Partially Ordered Sets ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Type Result ◽  
Ramsey Theorem ◽  
Partially Ordered ◽  
Ramsey Type

In this note we consider a Ramsey-type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao (2017).

scholarly journals Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials

10.37236/7686 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Miguel A. Méndez ◽  
Rafael Sánchez Lamoneda
Keyword(s):  
Algebraic Structure ◽  
Partially Ordered Sets ◽  
Koszul Duality ◽  
Ordered Sets ◽  
Algebraic Construction ◽  
Partially Ordered ◽  
Whitney Numbers ◽  
The Family ◽  
Sheffer Sequences ◽  
Sheffer Polynomials

We introduce a new algebraic construction, monop, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the enriched partition posets associated to a cancellative operad, and the subset posets associated to a cancellative monoid. Their Whitney numbers of the first and second kind are the connecting coefficients of two umbral inverse Sheffer sequences with the family of powers $\{x^n\}_{n=0}^{\infty}$. Equivalently, the entries of a Riordan matrix and its inverse. This aticle is the first part of a program in progress to develop a theory of Koszul duality for monops.

scholarly journals A Family of Partially Ordered Sets with Small Balance Constant

10.37236/7337 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Evan Chen
Keyword(s):  
Partially Ordered Sets ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Finite Poset ◽  
A Chain

Given a finite poset $\mathcal P$ and two distinct elements $x$ and $y$, we let $\operatorname{pr}_{\mathcal P}(x \prec y)$ denote the fraction of linear extensions of $\mathcal P$ in which $x$ precedes $y$. The balance constant $\delta(\mathcal P)$ of $\mathcal P$ is then defined by \[ \delta(\mathcal P) = \max_{x \neq y \in \mathcal P} \min \left\{ \operatorname{pr}_{\mathcal P}(x \prec y), \operatorname{pr}_{\mathcal P}(y \prec x) \right\}. \] The $1/3$-$2/3$ conjecture asserts that $\delta(\mathcal P) \ge \frac13$ whenever $\mathcal P$ is not a chain, but except from certain trivial examples it is not known when equality occurs, or even if balance constants can approach $1/3$.In this paper we make some progress on the conjecture by exhibiting a sequence of posets with balance constants approaching $\frac{1}{32}(93-\sqrt{6697}) \approx 0.3488999$, answering a question of Brightwell. These provide smaller balance constants than any other known nontrivial family.

On Operations and Linear Extensions of Well Partially Ordered Sets

Order ◽  
2004 ◽  
Vol 21 (1) ◽  
pp. 7-17 ◽  
Author(s):  
Maciej Malicki ◽  
Aleksander Rutkowski
Keyword(s):  
Partially Ordered Sets ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Partially Ordered
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