**7**

votes

**1**answer

98 views

### Equalizing Geometric means of Graph Cycles

Consider a strongly connected directed graph $G$. I have been stuck on the following question: can you assign real numbers in $[0,1]$ to each edge of $G$ so that the geometric mean of all cycles are ...

**2**

votes

**3**answers

396 views

### The number of submodules of $\mathbb{Z}_q^n$

Observe $\mathbb{Z}_q^n = \mathbb{Z}_q \times \cdots \times\mathbb{Z}_q$ as a module over $\mathbb{Z}_q\equiv\mathbb{Z}/q\mathbb{Z}$, for general $q$.
I am interested in the following questions:
How ...

**9**

votes

**1**answer

163 views

### A Graph-Theory Related Question

Let $n$ be a positive integer and partition a grid of $4n$ by $4n$ unit squares into $4n^2$ squares of sidelength $2$. (The squares with sidelength $2$ have all of their sides on the gridlines of the ...

**0**

votes

**0**answers

15 views

### Is there a shelling of a (threshold)shifted complex, such that any partial shelling is still (threshold)shifted?

first the relevant definitions:
A complex $\Delta \subset 2^{[n]}$ is a family of subsets of $[n]$ that is closed downwards, i.e. if $A \subset B$ and $B \in \Delta$, then $A \in \Delta$.
A complex ...

**4**

votes

**2**answers

256 views

### A property of uncountable almost disjoint families

Let $\mathcal{A}$ be an uncountable almost disjoint family (not necessarily maximal) of infinite subsets of $\mathbb{N}$. Denote by $\mathcal{A}_{\subseteq}=\{ B\subseteq\mathbb{N}:|B|=\omega \wedge ...

**5**

votes

**0**answers

61 views

### Upper bound for a sum including Andre polynomial coefficients

Let $ c_{n,k} $ be a sequence defined by the following relations: $\displaystyle c_{n,0} = 1, \hspace{0.1cm} (n \ge 1)$;
$$ c_{n,k} = (k+1) c_{n-1,k} +(n-2k+1) c_{n-1,k-1}, \hspace{0.5cm} (1 \leq k ...

**8**

votes

**0**answers

147 views

### An abstract zero-sum problem

I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...

**6**

votes

**1**answer

135 views

### Eigenvalue inequality for regular graphs

I recently proved an inequality relating some of the eigenvalues of a regular graph with each other, and I was wondering if it is already known. I was unable to find it online, and a quick skim ...

**3**

votes

**1**answer

104 views

### Algorithm to count the number of perfect matchings in non planar graph

I need to count the number of perfect matchings of a certain family of graphs. This family of graph is non planar and a type of snark. For the initial cases, it seems that this number is growing ...

**1**

vote

**0**answers

51 views

### On low rank $0/1$ real matrices with one connected component

Let $M$ be an $n\times n$ $0-1$ matrix of real rank $r$.
Moreover assume every row of $M$ is distinct and every column of $M$ is distinct. Such a matrix cannot be bigger than $2^r\times 2^r$.
As in ...

**2**

votes

**0**answers

62 views

### Number of multipartite partitions with odd components

For some positive integer $r$, by an $r$-vector I will mean an $r$-tuple $(a_1,a_2,\dots,a_r)$ with $a_1,\dots,a_r$ nonnegative integers not all zero, and I will call it odd if $a_1,\dots,a_r$ are all ...

**-8**

votes

**0**answers

50 views

### connectedness in Euclidean Space [closed]

f:[0,1] to R is a map defined by f(x)=sin(1/x),for x is not zero and
f(0)=0.X={(a,f(a))|a is in [0,1]}.prove that,X is a connected subset of
R^2 where the metric is usual euclidean metric on R^2.

**-6**

votes

**0**answers

66 views

### Continuity of Real line [closed]

f:R to R such that f attains every value exactly twice i.e. for all a in
R, {x in R|f(x)=a} is either empty or doubleton set.prove that,f is
discontinuous at infinitely many points .

**5**

votes

**1**answer

224 views

### A remarkable sum over partitions

While studying some seemingly unrelated topological questions, I have experimentally discovered what appears (to me) to be a remarkable sum over partitions. I was wondering if anyone knows how to ...

**3**

votes

**0**answers

59 views

### Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...

**2**

votes

**0**answers

95 views

### Zero-one links: how many, and how to produce?

For $m \geq 1$, define a link to be a zero-one word $w=d_0d_1 \ldots d_k$, where $d_0=0$ and $k=2^m-1$ , such that the words
$$ w_0=0^{m-1}d_0, w_1=w_0d_1, w_2=w_1d_2, \ldots, w_k = w_{k-1}d_k $$
...

**7**

votes

**1**answer

182 views

### Unusual isoperimetry and maximizing the measure of unions of translates of a set

Let me state a standard result first. Let a $A\subset \mathbb{R}^d$ be a set of fixed volume. Define $A_t$ to be the set of all points at distance at most $t$ from $A$. Then the volume of $A_t$ is ...

**0**

votes

**0**answers

49 views

### Polynomial time approximation schemes

The relationship between the minimum vertex cover and maximum independent set is a well established one. I was wondering if
If for some class of graphs $\mathcal{S}$ there exists a PTAS or even an ...

**3**

votes

**2**answers

88 views

### What are bounds for the number of monotone functions $M:P\rightarrow T$ where $P$ is a finite poset and $T$ is a finite totally ordered set?

For the case where $P=\{0,1\}^n$ and $T=\{0,1\}$ the number of such functions is called the $n$-th Dedekind number and I discovered that there is large literature on determining bounds for these ...

**11**

votes

**1**answer

565 views

### Where did the term “additive energy” originate?

A fundamental object in modern additive combinatorics and harmonic analysis is additive energy. Given a subset $A$ of (say) an abelian group $G$ the additive energy of $A$ is defined to be the ...

**-4**

votes

**0**answers

47 views

### Beginners level combinatorics: Using Binomial Coefficient in poker [closed]

I'm trying to learn about combinatorics, and since I enjoy poker I'm trying to use that to learn some mathematics.
I've been doing some calculations using the Binomial Coefficien but I can't get the ...

**4**

votes

**1**answer

105 views

### Coefficients of Ehrhart polynomials, in the binomial-coefficient basis

Let $P$ be the convex hull of a finite set of points in $\mathbb Z^d$, and $p(n) = \#\{nP \cap \mathbb Z^d\}$ be its Ehrhart polynomial, which is also the Hilbert polynomial of the corresponding ...

**-3**

votes

**0**answers

19 views

### Combinatorial problem - 6 classic dices [migrated]

can anyone help me with this task?
We throw with 6 classic(cube) dices. Two are yellow, two are blue and two are green. After every throw we put them into a line. How many different lines exist if we ...

**5**

votes

**1**answer

238 views

### Generating function for number of different tessellation checkered rectangle

Let $R_n$ be checkered rectangle sized $n \times 4, n \ge 1$.
Let $a_n$ be number of different $R_n$ tiling with rectangles sized $1 \times 3$.
$\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ ...

**-1**

votes

**0**answers

76 views

### What the number of the components of a specific subgraph? [closed]

Given a n vertices graph $G$, take two edge-disjoint matchings in $G$, namely $M_{1}$ and $M_{2}$, such that they cover $n-\alpha$ vertices each. In our case, $\alpha$ can be a constant or a function ...

**5**

votes

**0**answers

65 views

### What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...

**3**

votes

**1**answer

230 views

### Is there a name for this fast growing functions?

Define $F(n,i)=\prod_{j=1}^nj^{j^i}$.
$F(n,0)=n!$.
$F(n,1)$ is hyperfactorial.
Is there a term for $F(n,i)$? How fast do these grow? Is growth rate $2^{\frac{c_in^{i+1}\log_2n}{i+1}}$ with some ...

**12**

votes

**0**answers

131 views

### Coefficients of universal Schubert polynomials

Let $e_i^j$ be the elementary symmetric polynomial in $x_1,x_2,\ldots,x_j$. Then the ordinary Schubert polynomial has an expansion of the form
...

**11**

votes

**0**answers

146 views

### Is combinatorial automorphism of symmetric convex polytope always odd?

The question is formulated in the title. More precisely, if $P$ is an origin-symmetric convex polytope in $\mathbb{R}^d$, and $f$ is a bijective transform of the set of the vertices of $P$, which ...

**0**

votes

**0**answers

104 views

### The smallest disk containing all cirular arcs

In a comment to my recent question about covering segments by a disk, Gerhard Paseman has suggested a generalisation: replacing the segments of the original $n$-gon by a simple closed (say, convex) ...

**2**

votes

**1**answer

97 views

### Conceptual explanation for multiplicativity of theta generalization of extreme characters of U(infty)

A character of $U(\infty)$ is a continuous, positive definite class function $\chi: U(\infty) \longrightarrow \mathbb{C}$, which is normalized by $\chi(e) = 1$. Observe that the set of characters of ...

**11**

votes

**1**answer

140 views

### Graphs with a coloring that majorizes all other colorings

By a coloring of a graph $G = (V,E)$ I mean a map $\kappa:V\to\mathbb{N}$ such that $\kappa(u)\ne \kappa(v)$ whenever $u$ and $v$ are adjacent. (Sometimes this is called a proper coloring but I am ...

**2**

votes

**1**answer

126 views

### Alternating sign binomial identity [closed]

I recently noticed that for a triple of integers $k \geq 2$, $k \geq m \geq t \geq 1$, the following identity seems to hold
$\sum_{j=0}^{m-t} (-1)^{m-t-j}{k \choose j}{m-1-j \choose t-1}={k-t \choose ...

**4**

votes

**1**answer

110 views

### Demazure product in Coxeter and Artin groups

As a follow-up of Allen's question Coxeter exchanges in non-reduced words, I wonder whether it is known that the Demazure product is well-defined in Artin groups. This is:
Let $(W,S)$ be a Coxeter ...

**-1**

votes

**0**answers

35 views

### Categorification (Visually) of a Proset $A^*$ of All $(a_i)_{i\geq1}$ Over $A$ [closed]

(Mis)Understandingly, I can take that a subset of $A^*$ could be some infinite sequence $A_n=(a_i)_{i\geq1}$ with a preorder $\preceq$ between every other infinite sequence within $A^*$ creating an ...

**0**

votes

**0**answers

72 views

### Partial sum of binomial coefficients

For some integer $z \ge 2$ and large integer $n$ and $ t=\lceil \log n\rceil $, what is an approximate value for the following partial binomial sum?
$$ \sum_{i=0}^{n-t} \binom{n}{i}z^i .$$
Another ...

**6**

votes

**1**answer

328 views

### A property of the derivatives of a function

Suppose that $f,g_1,g_2,\dots$ are functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f'=f\,g_1$ and $g'_j=g_j^2-g_j g_{j+1}$. Here and in what follows, $j$ is any natural number. Then, by ...

**12**

votes

**0**answers

151 views

### Do the Laver tables converge to the Sierpinski triangle with a line segment sticking out in the hyperspace topology?

Let $(\{1,...,2^{n}\},*_{n})$ denote the $n$-th Laver table.
Let
$$C_{n}=\{(\frac{x}{2^{n}},\frac{x*_{n}y}{2^{n}})|x,y\in\{1,2,3,...,2^{n}\}\}$$
for all $n\in\mathbb{N}$.
Then since $C_{n}$ is a ...

**5**

votes

**1**answer

126 views

### Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices)

A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.
It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ...

**11**

votes

**1**answer

244 views

### Pursuit-Evasion type game on graph (“Flyswatter game”)

An instance of the "flyswatter game" is defined by a graph $G$ and positive integer $k$. There are two players, A (the 'fly') and B (the 'swatter'). Essentially, the fly moves around $G$ and the ...

**13**

votes

**2**answers

298 views

### Coxeter exchanges in non-reduced words

Let $Q$ be a word in the generators of some Coxeter group, and consider a subword $R$ (not necessarily reduced, though I might want $Q$ to be).
Define the greedy or Demazure product of $R$ as ...

**0**

votes

**1**answer

49 views

### The cost function in the Weighted Bipartite Matching Problem (a.k.a the Assignment Problem)

In the definition of this problem, the weight/cost function generally takes value in $\mathbb{Z}$ (or sometimes $\mathbb{Q}$).
This is what I observed from some books (e.g. "Combinatorial ...

**3**

votes

**1**answer

72 views

### Probabilistic many-to-one matching

Let $p<1$ be a constant. Consider two sets $A,B$ with $n$ and $nf(n)$ vertices, respectively, where $f(n)$ is an integer. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears ...

**2**

votes

**2**answers

108 views

### Matching with probabilistic edges

Let $p<1$ be a constant. Consider two sets $A,B$, each with $n$ vertices. For each pair $(a,b)\in A\times B$, the edge between $a$ and $b$ appears with probability $p$, independently of the ...

**1**

vote

**0**answers

95 views

### Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$

The problem may be formulated as follows:
We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...

**4**

votes

**4**answers

182 views

### Selecting subsets with size $\frac{n}{2}$ covering every pair of the elements

Given a set $S$ of $n$ elements. Let $T$ be the set of all subsets of $S$, with size $\frac{n}{2}$ ($n$ is even). We want to select a subset $T'$ of $T$, with the property that for any pair of the ...

**3**

votes

**1**answer

126 views

### Volume of the convex hull of the set of all graphic sequences of a given length

Consider the set of all graphic sequences with $n$ elements as a subset of $\mathbb{R}^{n}$, namely let
$$D(n)=\{(d_{1},\dots,d_{n})\in\mathbb{Z}_{+}^{n}:d_{1}\geq\dots\geq d_{n},\ ...

**7**

votes

**1**answer

261 views

### Linear relations among permutation matrices

Given a permutation $\sigma\in S_n$, let $P_\sigma$ denote the corresponding $n\times n$ permutation matrix. It is easy to see that for $n=3$, there is only one linear relation up to scaling given by ...

**1**

vote

**1**answer

88 views

### Eulerian graphs with prescribed number of edges

Under what conditions there exists an $n$-vertex eulerian graph with $m$ edges for $1\leq m\leq\frac{n(n-1)}{2}$?

**8**

votes

**1**answer

155 views

### Expansion in strongly regular graphs

Have you seen the following statement proven anywhere?
Let $G$ be a strongly regular graph with parameters $(n,k,\lambda,\mu)$ with $\lambda,\mu>0$. Then there is no set $A$ of at least $n/4$ ...