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

learn more… | top users | synonyms (1)

0
votes
0answers
45 views

Combinatorial approach to binomial sum identity? [on hold]

So I'm trying to prove the following: $$ \sum\limits_{k=0}^{n} {k \choose a} {n-k \choose b} = {n+1 \choose a+b+1}$$ And I'm a little caught on how to get started. It doesn't seem like a ...
3
votes
0answers
60 views

On a problem of sphere-packing for Reed-Solomon codes

Suppose we have an $[n, k+1, n-k]$ Reed Solomon code $\mathcal C$ over $\mathbb F_q$, where $n-k=d$ is the minimum distance, and suppose that $d=2t+1$. We know that for every $r \in \mathbb F_q^n$ the ...
0
votes
0answers
59 views

Hilbert-Poincaré series of a polynomial ring with non standard grading

Let us suppose we have as graded algebra $V=\mathbb{C}[z_0, ..., z_n]$ with $\deg z_0=a_0, \ldots, \deg z_n=a_n$, where we can suppose that not all the $a_i’$s are equal to 1. We know that we can ...
5
votes
0answers
56 views

Why have most maximal cliques of Paley graphs odd size?

I ask this question mainly by curiosity. See here for definitions and a plot of the clique numbers of the Paley graphs for the primes $p\equiv 1 \pmod 4$ up to $10000$. Is there an ...
0
votes
0answers
68 views

Computing the maximum of a (Classic) function involving factorial [migrated]

I'm trying to compute the global maximum (or an upper bound that does not depends on x) for the expression $$1/x!\binom{x+n-1}{x},$$ as a function of x. Where $n$ in a positive integer parameter. An ...
6
votes
0answers
150 views

Non-trivial bounds for polynomials at a fixed point

Let $f$ be a polynomial of degree $d$. Of course $|f(z)|\sim C|z|^d$ as $|z|\rightarrow\infty$ but also, since any polynomial is completely determined by its values at any $d+1$ points, we may ask how ...
-1
votes
0answers
42 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 ...
9
votes
3answers
331 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 ...
1
vote
0answers
65 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 ...
7
votes
2answers
104 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 ...
0
votes
0answers
42 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 ...
2
votes
1answer
63 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
3answers
328 views
+150

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 ...
9
votes
0answers
99 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 ...
0
votes
1answer
52 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 ...
1
vote
2answers
120 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 ...
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
270 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
599 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
179 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 ...
13
votes
1answer
220 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 ...
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. ...
13
votes
1answer
307 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 ...
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 ...
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 ...
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 ...
-3
votes
1answer
116 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 ...
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 ...
5
votes
0answers
138 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 ...
12
votes
2answers
460 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 ...
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 ...
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 ...
2
votes
0answers
92 views

reference on aperiodicity and cluster [closed]

From this image: I believe there is a message relating those clusters drawn in picture and aperiodic tiling. Does anyone have some reference on this? Thank you :)
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 $(a_1,a_2\dots a_n)$ ...
6
votes
1answer
147 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 ...
2
votes
0answers
74 views

second smallest eigenvalue Laplacian - submodular set function

Let $G$ be a connected unweighted undirected graph. In addition, let $\lambda_2(L(G))$ be the second smallest eigenvalue of the Laplacian matrix of graph $G$. Is $\lambda_2(L(G))$ a submodular set ...
1
vote
2answers
163 views

Cubic Cayley (undirected) graphs

The generators of a cubic Cayley graph always include at least one involution, since 3 is odd. There are then two possibilties for the other two generators: either (A) they are also (distinct) ...
3
votes
1answer
221 views

Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*} f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...
2
votes
0answers
54 views

Multi-dimensional permanent of structured tensor

I am facing the multidimensional permanent \begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation} of a 3-tensor $W_{j,k,l}$ of ...
1
vote
1answer
77 views

dual (p,q)-property

If the set system $(X,S)$ has the $(p,q)$-property does its dual system also have the property? (Possibly, for different $p$ and $q$.) Explicitly, I am asking about the equivalence of the following ...
4
votes
1answer
105 views

Maximum sets of lattice points such that only a few points collinear

Consider all the integer points $\in [0,n]\times[0,n]$, I want to find the maximum subset $S$ of which such that there are at most $n^\varepsilon(0<\varepsilon<1)$ points in $S$ collinear. So, ...
4
votes
2answers
130 views

Minimum number of transitive paths in tournament

Let $T$ be a tournament with $n$ vertices (i.e., between every pair of vertices there exists an edge in exactly one direction.) For any $k$, the vertices $A_1,A_2,...,A_k$ form a transitive path if ...
5
votes
0answers
124 views

Operator connected with Hermite polynomials

For $n \geq 1$, define the following operator $M_n$ on the ring of all polynomials with real coefficients. $$M_n P(x) = nP(x)^2 - x \int_0^x (P'(t))^2 \, \mathrm{d}t$$ Monomials $x^k$ are mapped to $n ...
2
votes
2answers
66 views

Largest eigenvalue adjacency matrix-link deletion

Let G be a connected undirected graph and G\e be a graph obtained by removing a random link e from the graph G. Let $\lambda_1(A(G))$ be the largest eigenvalue of the adjacency matrix of graph G. Is ...
1
vote
0answers
46 views

Minimal density hitting set for k-length arithmetic progressions

This is problem which came up in the process of designing a game. Thus, I don't know any previous work relevant to the problem. Fix a small set $D$ of a natural numbers. For example, $D=\{1,2,3\}$. ...
1
vote
0answers
94 views

Kempe chain color swaps in a partially colored map

Crossposted from math.stackexchange.com: http://math.stackexchange.com/questions/904932/kempe-chain-color-swaps-in-a-partially-colored-map Question: In this partially Tait's colored map, using ...
8
votes
2answers
325 views

Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?
2
votes
1answer
75 views

When does a hypergraph represent maximal independent sets?

Let $G = (V,E)$ be a simple graph. Then, we can view the set of maximal independent sets (or the set of maximal cliques) as a hypergraph $H = (V, E')$. This is quite a useful device when connecting ...