Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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6
votes
2answers
189 views

Square filling self avoiding walk

I want to create an algorithm that fills a square grid with a random Hamiltonian path starting at a particular point. See this example. One approach is to try a free direction as a next step, and ...
5
votes
1answer
60 views

Trees with a maximal convex hull: are the only optimal solutions Steiner trees?

For given $n\geqslant 3$, I'm looking for a connected set composed of $n$ equal segments in the plane such that the convex hull of it has maximal area $A(n)$. To simplify notation, we'll take ...
2
votes
1answer
43 views

Existence of neighborhood inclusion for 4-chordal graphs

Let $N(v)$ be the (open) neighbourhood set of a vertex $v$, and let $N[v]$ be the closed neighbourhood set of $v$. A graph $G$ is called 4-chordal if $G$ has no induced cycle with five or more ...
1
vote
0answers
28 views

How to distribute groups over activities in rounds [on hold]

This problem started with my sister asking me how she can distribute 12 groups over 6 activities in 6 rounds. She wants to organize a camp for for year students. There are 6 acitvities and in every ...
1
vote
3answers
176 views

a colouring / matching problem

While trying to find a bijection which preserves various combinatorial statistics, I was led to the problem below. Very much to my surprise, a closely related question, Coloring summands of given ...
1
vote
0answers
45 views

Expanding graphs from iterated zig-zag product

Let $\Gamma \, zz\, G$ denote the zig-zag product of graphs $\Gamma$ and $G$. Reingold, Vadhan, and Wigderson proved that the second largest eigenvalue of the normalized adjacency matrix of the ...
0
votes
0answers
39 views

Linear extensions and directed, rooted spanning trees

Let $P$ be a poset with a unique bottom element $\perp$ and view its Hasse diagram $H$ as an oriented graph. Can the principle of exclusion-inclusion be used to calculate the number of linear ...
7
votes
2answers
321 views

Quickly finding optimal subset of pairs of numerator and denominator terms for special objective functions

Given a multi-set of pairs $((a_i,b_i))_{i \in Y}$ of positive numerator and denominator terms (i.e. each pair has one numerator term and one denominator term), my general problem is to find the ...
8
votes
0answers
96 views

Computing exact or asymptotics for number of strings over an alphabet of size $n$ that have no non-trivial substrings that appear more than once

I ran across a seemingly relatively simple combinatorics problem that appears open. For an alphabet of size $n$, let $A(n)$ be the number of strings over the alphabet that have no substring of length ...
34
votes
0answers
720 views

Intersecting Family of Triangulations

Let $\cal T_n$ be the family of all triangulations on an $n$-gon using $(n-3)$ non-intersecting diagonals. The number of triangulations in $\cal T_n$ is $C_{n-2}$ the $(n-2)$th Catalan number. Let ...
1
vote
2answers
118 views

Non-DS circulant graphs

Let $p$ be a prime number greater than or equal to 11. Are there any cospectral non-isomorphic graphs with circulant graphs on $p$ vertices. Which circulant graphs over prime number of vertices ...
0
votes
1answer
51 views

Monotonicity of the gap of permutated sequence

Let $a$ be an arbitrary sequence and denote by $\mbox{gap}_k(a) = a_{(k)} - a_{(k+1)}$, where $a_{(k)}$ is the $k$th largest component of $a$. Of course, $k+1$ should be no larger than the length of ...
2
votes
1answer
119 views

Vanishing homology of simplicial complexes with few facets

Let $K$ be a simplicial complex with $n$ vertices and $n-t$ facets, where $t \geq 3$. Is it true that the $(n-3-j)$th reduced homology (with coefficients in a field) of $K$ vanishes for $0 \leq j ...
5
votes
2answers
268 views

Random Vornoi Diagrams (particular measures)

This is my second question about Random Voronoi diagrams, in my first question was given some excellent advice but i was not clear in explaining what i was looking for. I'm interested to know ...
8
votes
2answers
595 views

Random Voronoi Diagrams

I'm interested in what research has already been done with regards to the statistics of random voronoi diagrams. I have had a look on google scholar and results are a little inconclusive. I'm ...
-2
votes
0answers
43 views

Derangement and inclusion -exclusion [closed]

I understand the recursive formula for derangement but don't understand how and why inclusion-exclusion principle is used for the proof of derangement formula.I want to know the intuitive explanation ...
2
votes
1answer
177 views

On the maximum number of $t$-subset of $\{1,\ldots, n\}$ having pairwise singleton or empty intersections

Let $n$, $t \in \mathbb N$ two natural numbers such that $0<t<n$, and let $A$ be a set of $n$ elements. We call a quasi-partition or q-p of $A$ a subset $W \subset \mathcal P(A)$ such that we ...
1
vote
1answer
408 views

A generalized urn-ball matching problem; Complicated combinatoric/probabilistic limit

I'm looking for a generalization to the urn-ball matching problem. As a reminder of what I've got in mind, here's the simple version: Randomly assign (with replacement) $N$ balls to $M$ urns. ...
13
votes
1answer
215 views

Flag complexes that are shellable but not vertex decomposable

As the title suggests, I was wondering if anyone can point me to any examples in the literature to flag complexes that are shellable but not vertex decomposable. It is well-known that if a ...
2
votes
1answer
164 views

Cospectrality and dimension of graphs

Firstly, I apologize if the question is long. I appreciate any helpful answers and ideas. In the following all graphs are simple and connected. Let $G$ be graph with vertex set ...
7
votes
3answers
685 views

A conjecture about the entropy of matrix vector products

Consider a random $m$ by $n$ partial circulant matrix $M$ whose entries are chosen independently and uniformly from $\{0,1\}$ and let $m < n$. Now consider a random $n$ dimensional vector $v$ ...
14
votes
1answer
403 views

3D generalizations of permutations, RSK correspondence, contingency tables, etc.

I want to gather facts and questions related to 3D generalizations of permutations, RSK correspondence, contingency tables, etc. One reason I am interested in this is because it is potentially related ...
12
votes
1answer
306 views

A method for making a graph bipartite

Given any graph $G$, can we find a bipartite subgraph of $G$ with at least $e(G)/2$ edges ($e(G)$ is the number of edges in $G$) by sequentially deleting the edge belonging to the most number of odd ...
39
votes
4answers
6k views

Do there exist chess positions that require exponentially many moves to reach?

By "chess" here I mean chess played on an $n\times n$ board with an unbounded number of (non-king) pieces. Some care is needed if you want to generalize some of the subtler rules of chess to an ...
6
votes
1answer
142 views

Counting cyclic binary sequences of length $n$ where ones appear in blocks of length at least $k$

How many binary cyclic sequences of length $n$ exist, where ones only appear in blocks of length at least $k$? We do not consider sequences that result from each other by a cyclic shift equivalent. ...
1
vote
1answer
78 views

Number of points in an intersecting linear hypergraph

I first asked the question below at math.stackexchange.com ( http://math.stackexchange.com/questions/920442/number-of-points-in-an-intersecting-linear-hypergraph ) but somebody suggested I ask it in ...
6
votes
1answer
385 views

More asymptotics for trees

This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...
2
votes
0answers
62 views

More 3-connected cubic graphs with all 2-factors of same cycle type?

The setup is as in this question: Let $G$ be a 3-connected cubic graph. If all 2-factors of $G$ are isomorphic (as graphs), i.e. all have the same partition $\pi$ as cycle type, we'll say that $G$ is ...
1
vote
1answer
143 views

Name search for special Linear Integer Program

I am looking for a name for the following question in literature! (and if you know it, then it would be great) I couldn't find it and due to wide audience here, presumably you know more. Thank you ...
19
votes
1answer
605 views

Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
11
votes
1answer
410 views

Faster multiplication with a restricted set of multiplicands?

Let $A$ be a set of $k>1$ distinct elements from a semigroup. We wish to compute the product $$ p=b_1 b_2 \cdots b_n$$ where each $b_i\in A$. Clearly $n-1$ multiplications suffice to compute $p$; ...
12
votes
2answers
458 views

The sum of the carries when adding and multiplying two numbers in base p

Let $\sigma_p(m,n)$ (resp. $\pi_p(m,n)$) denote the sum of the carries when adding (resp. multiplying) the numbers $m=\sum_{k\ge0}m_kp^k$ and $n=\sum_{k\ge0}n_kp^k$ using base-$p$ arithmetic where ...
5
votes
1answer
290 views

Are there number-theoretic graphs that are far from being isomorphic

I say that two graphs $G_1=(V_1,E_1)$, $G_2 = (V_2,E_2)$ with the same number of vertices, edges, are $\epsilon$-far from being isomorphic, if for any bijection between $V_1$ and $V_2$, the fraction ...
5
votes
0answers
137 views

(Connected) Cayley graphs of PSL(2,q) from (2,3,n)-triples

Let $G = PSL(2,q)$. I'm interested in the Cayley graphs of $G$ generated by triples $(A,BAB^{-1},B^{-1}AB)$, where $A, B \in G$ are elements of order $2, 3$ respectively: such a triple generates all ...
5
votes
1answer
315 views

Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$. Consider the bipartite ...
-3
votes
1answer
115 views

How many sequences of length n satisfy these constraints? [closed]

I want to count the number of unique sequences of length n with the following constraints. Each element of the sequence is an integer in $\lbrace 1,2,\dots,n\rbrace$. Each two adjacent elements of ...
19
votes
1answer
787 views

Recognize this strange expression from linear algebra?

I've come across an odd-looking expression and oh how I wish I had a more elegant description of it. Maybe someone who enjoys symmetric bilinear forms in characteristic two will recognize it? Or ...
0
votes
1answer
84 views

Comparing ideals in posets

Consider a partially ordered set $P$, and two upper sets $U_1$, $U_2$ in this poset. What are some natural ways to measure how equal these two upper sets are? This question arise naturally in the ...
14
votes
1answer
593 views

Are There Always Group Generators Which Give Unimodal Growth?

Suppose $G$ is a $k$-generated finite group. Is there always a set of $k$ elements which generate the group and have a unimodal counting function? Background: The counting function, $f(n)$, is a ...
2
votes
1answer
156 views

A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
0
votes
1answer
182 views

When is a power of an indeterminate in an ideal with 2 generators?

If I have an ideal ${\frak a} \colon= (f(X,S), g(X,S))$ of height $2$ in ${\Bbb C}[X, S]$, is it easy to know what power of $S$ is contained in $\frak a$? For example, what is the minimal number $m$? ...
24
votes
6answers
3k views

What is known about this plethysm?

Let $S^{\lambda}$ be a Schur functor. Is there a known positive rule to compute the decomposition of $S^{\lambda}(\bigwedge^2 \mathbb{C}^n)$ into $GL_n(\mathbb{C})$ irreps? In response to ...
2
votes
1answer
186 views

Coloring vertices in a cubic lattice graph and counting edges between vertices of identical and vertices of distinct coloration

Take an $A \times B \times C$ cubic lattice graph $G$, and paint $k_1$ vertices with color $c_1$ & $k_2$ vertices with color $c_2$, where $(k_1 + k_2)$ is equal to the total vertex count. Let ...
0
votes
1answer
103 views

Collecting terms of a linear expression with nested sums and combinatorics in coefficients

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
9
votes
1answer
310 views

Coherence between different ranking methods of a graph's vertices

Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher. Two natural ways of doing it are: By the degrees. By the entries in a ...
4
votes
2answers
92 views

Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?

For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
3
votes
1answer
105 views

Uniformly permutation and the length of a size biased cycle

The cycle containing $1$ of a uniform permutation has length which is uniformly distributed. I was wondering if the converse is true: Suppose $\sigma$ is a permutation on $\{1,\dots,n\}$ and let ...
5
votes
1answer
403 views

Edge Covering Shortest path

I have a graph of Points G(V,E) and I want to find the shortest path covering all the edge, I want the minimum number of edge repetitions . Which is the best way to reduce this problem to well know ...
6
votes
1answer
145 views

Coloring summands of given n-partition with given weights of colors

Let $\lambda$ and $\sigma$ be partitions of $n$: $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$ Let $M_{\lambda \sigma}$ be the number of ways to colour the parts ...
0
votes
1answer
72 views

reference request for automata of this type [closed]

Consider the list of length $m$ $(1,0,\dots 0)$ we call this list $l_1$, we now define a sequence of lists recursively, where $l_1$ is the previous list, and if $l_n$ is the list $(a_1,a_2\dots a_n)$ ...