All Questions
Tagged with enumerative-combinatorics posets
10 questions
8
votes
1
answer
334
views
What is the Möbius function for the lattice of partial partitions?
Let $n$ be a positive integer. Let $P$ be the set of partitions of subsets of
$\{ 1, 2, \dotsc, n \}$ (so, for example, when $n=2$, the set $P$ contains $\emptyset$, $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{...
- 557
8
votes
0
answers
155
views
Partial order on graphs induced by homomorphism counts
For graphs $F$ and $G$, let $\hom(F,G)$ denote the number of homomorphisms (adjacency preserving maps) from $F$ to $G$. Define a relation $\le_{\hom}$ on (isomorphism classes of) graphs as $G \le_{\...
- 2,339
3
votes
0
answers
214
views
Does every finite lattice embed into a finite Eulerian lattice?
A finite Boolean lattice is a lattice isomorphic to the subset lattice of a finite set. Every Boolean lattice is Eulerian, namely, a graded lattice $L$ such that $\mu(a,b) = (-1)^{|b|-|a|}$ for all $a,...
7
votes
1
answer
424
views
Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history
Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces.
Counting the ...
3
votes
0
answers
179
views
A boolean representation of the Möbius function on a finite lattice
Let $(L,\wedge , \vee)$ be a finite lattice with minimum $\hat{0}$ and maximum $\hat{1}$.
Consider the Möbius function $\mu$ on $L$ defined inductively by $$\mu(\hat{1}) = 1 \text{ and } \mu(a) = - \...
2
votes
1
answer
83
views
Birkhoff Lattice of a forest
In my research, I stumbled upon a particular kind of poset and I was wondering, whether there is something in the literature (I could not find anything so far).
They are distributive lattice $L$ ...
- 243
7
votes
1
answer
169
views
Enumerative characterisation of boolean lattices II
This is a sequel of this post.
The boolean lattice $B_n$ is graded with rank numbers $\binom{n}{0}, \binom{n}{1}, \dots, \binom{n}{n}$, and $n2^{n-1}$ edges.
Question: Is a graded lattice with the ...
5
votes
1
answer
247
views
Enumerative characterisation of boolean lattices
The boolean lattice of rank $n$ (noted $B_n$) is the subset lattice of $\{1,2, \dots , n \}$.
See the Hasse diagram of $B_3$ below:
The Hasse diagram of $B_n$ is of length $n$, with $2^n$ vertices ...
12
votes
2
answers
758
views
Principal Order Ideals in the Weak Bruhat Order
Let $\sigma\in S_n$ be a permutation on $n$ elements, and $\mathrm{Inv}(\sigma):=\{(i,j) : 1\leq i<j\leq n\text{ and }\sigma(i)>\sigma(j)\}$ be its set of inversions. In the weak order on ...
- 1,324
6
votes
0
answers
207
views
When can we determine an $f$-vector or rank-generating function from its unordered list of coefficients?
Let $f_i$ be the number of $i$-dimensional faces in a $d$-dimensional simplicial complex $\Delta $. Recall that the $f$-vector of $\Delta $ is the vector $(f_{-1},f_0,f_1,\dots ,f_d)$ where $f_{-1}=...
- 3,532