Order Preserving Maps and Linear Extensions of a Finite Poset

1985 ◽  
Vol 6 (4) ◽  
pp. 738-748 ◽  
Author(s):  
David E. Daykin ◽  
Jacqueline W. Daykin
Keyword(s):  
Linear Extensions ◽  
Finite Poset

scholarly journals Promotion and Evacuation

10.37236/75 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Vector Space ◽  
Hecke Algebra ◽  
Linear Extensions ◽  
Linear Transformations ◽  
Basic Properties ◽  
Finite Poset ◽  
Order Ideals ◽  
Maximal Chains

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

scholarly journals Fixed Points of the Evacuation of Maximal Chains on Fuss Shapes

10.37236/6898 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Hsiang-Chun Hsu ◽  
Yu-Pei Huang
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Rectangular Shape ◽  
Explicit Characterization ◽  
Finite Poset ◽  
Maximal Chains

For a partition $\lambda$ of an integer, we associate $\lambda$ with a slender poset $P$ the Hasse diagram of which resembles the Ferrers diagram of $\lambda$. Let $X$ be the set of maximal chains of $P$. We consider Stanley's involution $\epsilon:X\rightarrow X$, which is extended from Schützenberger's evacuation on linear extensions of a finite poset. We present an explicit characterization of the fixed points of the map $\epsilon:X\rightarrow X$ when $\lambda$ is a stretched staircase or a rectangular shape. Unexpectedly, the fixed points have a nice structure, i.e., a fixed point can be decomposed in half into two chains such that the first half and the second half are the evacuation of each other. As a consequence, we prove anew Stembridge's $q=-1$ phenomenon for the maximal chains of $P$ under the involution $\epsilon$ for the restricted shapes.

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.

scholarly journals Connected Order Ideals and $P$-Partitions

10.37236/6463 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Ben P. Zhou
Keyword(s):  
Rational Functions ◽  
Independent Sets ◽  
Simple Graph ◽  
Nonempty Intersection ◽  
Alternative Proof ◽  
Linear Extensions ◽  
Connected Component ◽  
Finite Poset ◽  
Set Inclusion ◽  
Order Ideals

Given a finite poset $P$, we associate a simple graph denoted by $G_P$ with all connected order ideals of $P$ as vertices, and two vertices are adjacent if and only if they have nonempty intersection and are incomparable with respect to set inclusion. We establish a bijection between the set of maximum independent sets of $G_P$ and the set of $P$-forests, introduced by Feray and Reiner in their study of the fundamental generating function $F_P(\textbf{x})$ associated with $P$-partitions. Based on this bijection, in the cases when $P$ is naturally labeled we show that $F_P(\textbf{x})$ can factorise, such that each factor is a summation of rational functions determined by maximum independent sets of a connected component of $G_P$. This approach enables us to give an alternative proof for Feray and Reiner's nice formula of $F_P(\textbf{x})$ for the case of $P$ being a naturally labeled forest with duplications. Another consequence of our result is a product formula to compute the number of linear extensions of $P$.

Relative linear extensions of sextic del Pezzo fibrations over curves

2020 ◽  
Vol 21 (2) ◽  
pp. 1371-1409 ◽  
Author(s):  
Takeru Fukuoka
Keyword(s):  
Linear Extensions ◽  
Del Pezzo

scholarly journals A Sequential Importance Sampling Algorithm for Counting Linear Extensions

2020 ◽  
Vol 25 ◽  
pp. 1-14
Author(s):  
Alathea Jensen ◽  
Isabel Beichl
Keyword(s):  
Importance Sampling ◽  
Sequential Importance Sampling ◽  
Linear Extensions ◽  
Sampling Algorithm

scholarly journals Accidental Gauge Symmetries of Minkowski Spacetime in Teleparallel Theories

Universe ◽  
2021 ◽  
Vol 7 (5) ◽  
pp. 143
Author(s):  
Jose Beltrán Jiménez ◽  
Tomi S. Koivisto
Keyword(s):  
General Relativity ◽  
Minkowski Spacetime ◽  
Linear Extensions ◽  
Gauge Symmetries ◽  
Linear Spectrum ◽  
Massless Spin ◽  
Non Linear ◽  
Common Framework ◽  
Local Symmetries ◽  
Quadratic Action

In this paper, we provide a general framework for the construction of the Einstein frame within non-linear extensions of the teleparallel equivalents of General Relativity. These include the metric teleparallel and the symmetric teleparallel, but also the general teleparallel theories. We write the actions in a form where we separate the Einstein–Hilbert term, the conformal mode due to the non-linear nature of the theories (which is analogous to the extra degree of freedom in f(R) theories), and the sector that manifestly shows the dynamics arising from the breaking of local symmetries. This frame is then used to study the theories around the Minkowski background, and we show how all the non-linear extensions share the same quadratic action around Minkowski. As a matter of fact, we find that the gauge symmetries that are lost by going to the non-linear generalisations of the teleparallel General Relativity equivalents arise as accidental symmetries in the linear theory around Minkowski. Remarkably, we also find that the conformal mode can be absorbed into a Weyl rescaling of the metric at this order and, consequently, it disappears from the linear spectrum so only the usual massless spin 2 perturbation propagates. These findings unify in a common framework the known fact that no additional modes propagate on Minkowski backgrounds, and we can trace it back to the existence of accidental gauge symmetries of such a background.

Linear extensions of dynamic systems and the reductibility problem

1970 ◽  
Vol 8 (4) ◽  
pp. 722-728
Author(s):  
S. B. Katok
Keyword(s):  
Dynamic Systems ◽  
Linear Extensions

Existence of Green-Samoilenko Functions of Some Linear Extensions of Dynamical Systems

2004 ◽  
Vol 7 (4) ◽  
pp. 454-460
Author(s):  
H. M. Kulyk ◽  
V. L. Kulyk
Keyword(s):  
Dynamical Systems ◽  
Linear Extensions

scholarly journals Linear Extensions of N-free Orders

Order ◽  
2014 ◽  
Vol 32 (2) ◽  
pp. 147-155 ◽  
Author(s):  
Stefan Felsner ◽  
Thibault Manneville
Keyword(s):  
Linear Extensions
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