All Questions
19 questions
9
votes
1
answer
542
views
Reference request: number of antichains of a partially ordered set
Let $\mathbb{N}$ denote the set of all positive integers. For each $n \in \mathbb{N}$, define the set $$ P_n = \{ (a,b) \in \mathbb{N} \times \mathbb{N} : 1 \leq a \leq b \leq n \} $$ and consider the ...
- 563
5
votes
0
answers
201
views
Is this "trimming" of a supersolvable semimodular lattice known?
Let $L$ be a finite (upper) semimodular lattice. Recall that this means $L$ is graded and its rank function $\rho\colon L \to \mathbb{N}$ satisfies
$$ \rho(x) + \rho(y) \geq \rho(x\vee y)+\rho(x \...
- 24.2k
6
votes
1
answer
201
views
Order ideals of positive root systems and avoiding group elements in the Weyl group
Let $X$ be the poset of positive roots of a finite root system of Dynkin type $Q$.
Question 1: In Dynkin type $A_n$, is it true that the poset of order ideals of $X$ is isomorphic to the poset of [2,...
- 26.5k
10
votes
1
answer
464
views
A definition in poset theory
I am working on a article in poset theory. In that article, I am defining a subposet of a poset. The definition is following:
Let $P$ be a finite poset. A subposet $P'$ of $P$ is called closed under ...
- 103
3
votes
0
answers
123
views
About finite posets without intervals of size 3
Let $P$ be a finite poset (partially ordered set).
I am wondering whether the following condition on $P$ has been studied somewhere:
(#) No interval $[a,b]$ in $P$ has $3$ elements.
Note that ...
- 3,587
6
votes
1
answer
432
views
What is the Möbius function of substrings?
Define a poset on the set of all finite binary strings, defined by $a \le b$ whenever $b = uav$ for (possibly empty) binary strings $u, v$.
What is the Möbius function of this poset?
- 63
7
votes
0
answers
206
views
Classification of posets that are quotient posets of the Boolean lattice
Quotient posets of the Boolean lattice $B_n$ have interesting properties and are for example discussed in chapter 5 of Stanley's book on algebraic combinatorics.
$B_n/G$ for a subgroup $G$ of the ...
- 26.5k
10
votes
1
answer
407
views
Poset-troids …?
In many respects,
spanning tree : graph :: linear extension : poset
For instance, the number of spanning trees/linear extensions is a measure of the "richness" or "complexity" of the graph/poset. ...
- 24.2k
8
votes
1
answer
253
views
Finite posets for which all intervals are atomic
Let $P$ be a finite poset which is a lattice with $0,1 \in P$.
An atom in $P$ is an upper cover of $0$ and a coatom is a lower cover of $1$.
$P$ is atomic if every element is a join of atoms and ...
- 3,271
3
votes
0
answers
272
views
Reference request: Representing posets by integer divisibility
Does anyone know of an early published reference for the (very easy) fact that all finite posets can be represented as the poset of divisibility of a finite set of integers?
Page 1 of Birkhoff's ...
- 18.6k
1
vote
0
answers
116
views
Does this expression involving the order complex of a poset ring a bell?
Let $\mathcal{L}$ be a meet-semilattice, and denote by $\Delta(\mathcal{L})$ the poset of chains in $\mathcal{L}\setminus\{\hat 0\}$, where $\hat 0$ is the minimum element of $\mathcal{L}$.
Let $\...
- 2,788
0
votes
0
answers
131
views
terminology: monotone maps of posets such that the image of a lower set is a lower set
How are called in combinatorics
monotone maps of partially ordered sets such that the image of a lower set is a lower set, i.e. closed (or open) maps of finite topologies? Is there a classification ...
- 113
7
votes
1
answer
457
views
Unique factorization of posets
Given two finite posets $P$ and $Q$, we can form the direct product poset $P \times Q$ whose elements are pairs $(p,q) \in P \times Q$ with $(p,q) \leq (p',q')$ if $p \leq p'$ and $q \leq q'$. Let us ...
- 24.2k
10
votes
0
answers
191
views
Ideals in strong Bruhat order
Recall that a (lower) ideal in a poset $(P, \le)$ is a subset $I\subset P$ such that $x\in I \Rightarrow y\in I$ for all $y\le x$. In am interested in ideals in finite Coxeter groups $W$ equipped ...
- 31.2k
5
votes
1
answer
261
views
Is there a standard name for this poset
I've run into the following poset and I would expect it has a standard name. Let $n\geq k\geq 0$. Then $P_{n,k}$ consists of all $k$-element subsets of $\{1,\ldots,n\}$ ordered by $X\leq Y$ if $X=\{...
- 38.6k
5
votes
1
answer
177
views
Reference for statement that almost every $n$-element partial order has trivial automorphism group
I'm looking for a reference for the statement that almost every partial order on $n$ elements has trivial automorphism group. I've been told that this is a folklore result. Does anyone know of a ...
- 462
7
votes
0
answers
259
views
"Double convolution" with the Mobius function on a poset
Let $f$ and $g$ be arbitrary (say integer-valued) functions on some poset $P$, and say $\mu$ is the Mobius function of $P$. I'm studying a quantity that's a sort of "double convolution" of $f$ and $g$ ...
- 1,510
3
votes
0
answers
208
views
References for properties/examples of breadth in (semi)lattices
This is in some sense following up on my earlier question Is there existing terminology for this technical condition on semilattices? and the answer given by NN.
I am currently revising the paper ...
- 25.8k
3
votes
1
answer
361
views
Posets of finite sequences are highly connected
I need the following result for an example in a paper I'm writing. It's easy enough to prove, but I'd prefer to just give a reference. Does anyone know one?
Fix $1 \leq k \leq n$. Define $X_{n,k}$ ...
- 44.8k