**6**

votes

**0**answers

96 views

### Inequality among domino tilings of large triomino shapes

Inspired by this question, which asks for what shape maximizes the number of domino tilings, I want to ask the following seemingly simpler question, which I have been thinking about for a while:
...

**15**

votes

**5**answers

587 views

### The smallest disk containing all sides of an $n$-gon

Start with a regular $n$-gon of side 1 and consider its sides as open segments that can be moved around in the plane, allowing only translations. Two segments may not intersect.
What is the ...

**14**

votes

**2**answers

564 views

### A conjecture about $\lfloor n!\cdot q/e\rfloor-\,!n\cdot q$

I was thinking about this question asked at Math.SE, when I came up with the following conjecture.
For every $q\in\mathbb Q$ consider a sequence $s_n^{(q)}$ (terms within the sequence are indexed by ...

**3**

votes

**1**answer

102 views

### Vertex-connectivity of connected, vertex-transitive graphs without $K_4$ is maximum possible

A graph is said to have optimal vertex connectivity if its vertex connectivity equals its minimum degree. According to this arXiv preprint, it was shown by Mader in (Arch. Math., 1970) and (Math. ...

**0**

votes

**1**answer

107 views

### An asymptotic intersection problem

Given a set of $n\in\Bbb N$ integers $\mathcal S$ and $\mathsf s,\mathsf t\in\big(0,1\big)$, suppose we choose two sets:
$$\mathcal S_{\mathsf{1}}\subseteq\mathcal S$$
$$\mathcal ...

**0**

votes

**0**answers

39 views

### Is the laplacian spectral radius of a directed grpah a non-decreasing function?

Given a directed graph G and its respective laplacian matrix $L = D-A$ where $A$ is the adjacency matrix ($A_{ij}=1$ if there is a link from $j$ to $i$ and $A_{ij}=0$ otherwise) and D is the diagonal ...

**5**

votes

**1**answer

232 views

### $(n-2)$-blocking sets in $AG(n,2)$

Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.
I have seen a lot work related to minimal $(n-1)$-blockings set.
...

**4**

votes

**2**answers

310 views

### Can you simplify (or approximate) $\sum_{n=0}^{N-1}\begin{pmatrix}N-1\\n\end{pmatrix}\frac{(-1)^n}{n+1} e^{-\frac{n}{2(n+1)}\lambda}$?

Let $\begin{pmatrix}x\\y\end{pmatrix}$ be the binomial coefficient. I am trying to get a better understanding of the sum
\begin{equation}
...

**2**

votes

**3**answers

203 views

### Making integer multisets graphic

Let $M=(X,f)$ be a multiset, where $X$ is the underlying set of elements and $f:X\rightarrow\mathbb{N}$ is the multiplicity function. For every $k\in\mathbb{N}$ put $k\cdot M:=(X,k\cdot f)$. It is ...

**7**

votes

**1**answer

93 views

### When is the diagonal of a rational bivariate power series again rational

Given a rational bivariate power series $F(x,y)=\sum{a_{n,m}x^ny^m}$, the diagonal function $G(t):=\sum{a_{n,n}t^n}$ is known to be algebraic, although not rational in general. I was wondering if ...

**19**

votes

**0**answers

306 views

### Zero curves of Tutte Polynomials?

There is an extensive theory of the real and complex roots of the chromatic polynomial of a graph, a substantial fraction of this being due to the connections between the chromatic polynomial and a ...

**8**

votes

**1**answer

312 views

### Removing singularities in generating functions

This is a problem about the practicalities of removing singularities in multivariable complex functions.
In trying to derive the generating function (in two variables) for a certain problem in ...

**1**

vote

**2**answers

72 views

### Position likelihood in a 2D graph [closed]

I am looking for general principles or specific answers to this generic example.
Assume a 2d grid with no boundaries and a roving dot (ant/drunk guy/particle) that is initially located at some ...

**4**

votes

**0**answers

71 views

### A relaxation of proper coloring

I am wondering if the following relaxation of proper coloring appears somewhere. I have tried some searching and have found a few relaxations of proper coloring, but none the coincides with what I ...

**6**

votes

**2**answers

262 views

### extremal bipartite graph

I'm facing the following question:
Given a bipartite graph $G = (L \cup R, E)$.
Let $n = |L|$, $m = |R|$, and a parameter $k \in \mathbb{N}$, $n > m > k$.
What is a minimal possible number ...

**6**

votes

**1**answer

243 views

### Subsets of [1..N] with no three-term arithmetic progressions and no large gaps

Let S be a subset of [1..N] containing no three-term arithmetic progression, and let h(S) be the size of the largest gap between two consecutive elements of S. By Roth's theorem, h(S) has to grow ...

**4**

votes

**1**answer

87 views

### Is there a nice form for the Frobenius characteristic of a border shape character?

Let $\chi^V$ be the character of a border strip Specht module, i.e. a Specht
module for a skew tableau that contains no $2 \times 2$ square. I know that
the Frobenius characteristic of $\chi^V$ is ...

**2**

votes

**0**answers

131 views

### On sets of coprime numbers

We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$
Denote by ...

**2**

votes

**3**answers

294 views

### An identity involving a product of two binomial coefficients

I'm trying to find a closed formula (in the parameters $q,N$) for the following sum:
$$ \sum_{k=q}^{N} {{k-1}\choose{q-1}} {{k}\choose {q}} $$
Can anybody give me a lead?
Lior

**7**

votes

**2**answers

239 views

### A Different 2-factor in a graph

We know that a k-factor of G is a k-regular spanning subgraph of G. And if G is 4-regular (or 2k-regular), it can be partitioned into 2 (k) edge-disjoint 2-factors (Petersen 1891).
My question is in ...

**-1**

votes

**2**answers

73 views

### Is there a formula that determines the size of the leafage of a graph's spanning tree? [closed]

In general terms, all the spanning trees of a graph G have the same number of leaves.
Is there any formula that allows us to know the number of leaves in terms of |V| and |E| for any spanning tree of ...

**1**

vote

**1**answer

107 views

### Strongly connected graph and the eigenvalues of the laplacian matrix

Given a directed graph $G$, consider that $G$ is strongly connected iff every vertex $i$ in $G$ has inner degree $k_i\geq 1$. Reformulation of this definition: $G$ is strongly connected iff for any ...

**0**

votes

**1**answer

67 views

### An asymptotic set containment problem [closed]

Given a set of $n\in\Bbb N$ integers $\mathcal S$, suppose we choose two sets:
$$\mathcal S_{\mathsf{small}}\subseteq\mathcal S$$
$$\mathcal S_{\mathsf{big}}\subseteq\mathcal S$$ with cardinalities
...

**4**

votes

**1**answer

264 views

### Number of different positions of rooks on chessboard

I know that this topic as been mentioned before, but no accurate answer has been provided.
Suppose we have to place $n$ rooks on $n \times n$ chessboard so that no one attacks another. How to count ...

**7**

votes

**2**answers

254 views

### Distribution of $\max_{n \ge 0} S_n$, random walk

Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the ...

**0**

votes

**2**answers

106 views

### Infinite k-connected planar graphs

By planar I mean there is no $K_{3,3}$ minor of $K_5$ minor. Also, I am only considering the $\mathbb{R}^2$ surface, not a torus not any other surfaces.
I know that to construct such graph, For $k ...

**4**

votes

**1**answer

198 views

### Combinatorial description of a 120-cell

I'd like a combinatorial description of the 1-skeleton of the 120-cell (roughly) along the lines of the following description of the 1-skeleton of a dodecahedron.
(View all elements of product sets ...

**19**

votes

**1**answer

622 views

### Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$

When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following:
Problem. We have a surface of a cube $n\times n \times n$ such that each ...

**4**

votes

**1**answer

80 views

### Weights on cyclic orderings

Are there standard or known weights/metrics on cyclic orders?
Cyclic orderings are different ways of listing elements from a finite set, where you call two lists the same if they differ only by a ...

**2**

votes

**1**answer

130 views

### Does this version of Hadwiger's conjecture hold for graphs with infinite chromatic number?

Let $G = (V,E)$ be a finite, simple, undirected graph. Hadwiger's conjecture states that
(Hadw): $K_{\chi(G)}$ is a minor of $G$.
It turns out that for finite graphs, (Hadw) is equivalent to the ...

**4**

votes

**1**answer

120 views

### Existence of an infinite word with a predetermined asymptotic for the word complexity

Let $w$ be an infinite binary word, for example: $$1010100001 0010011000 0001001110 0101011011 \dots$$
Let $N_w(k)$ be the set of distinct subwords of $w$ of length $k$, and $n_w(k)$ the cardinal of ...

**1**

vote

**1**answer

50 views

### Summation of multinominal coefficients with extra bounds on summation indices

My question is related to the sum
\begin{equation}
S(n,N) = \sum_{k_1+k_2+...+k_N=n}\frac{n!}{(k_1!)\cdot(k_2!)\cdot...\cdot(k_N!)} = N^n,
\end{equation}
which is comes from the multinomial ...

**2**

votes

**0**answers

51 views

### The numbers of edge colorings and partitions into vertex covers

Let $G = \langle X \cup Y, E\rangle$ be a $d$-regular bipartite graph such that $|U|=|V|=n$, and assume that $d$ divides $n$. Let $F(G)$ denote the number of proper edge colorings of $G$ using $d$ ...

**3**

votes

**1**answer

90 views

### Number and asymptotic for cyclic sequences

Cyclic sequence is equivalence class of cyclic shift action.
If $a = (a_1, ... , a_i)_c$ is cyclic sequence then $(a_1, a_2, \ldots a_{i-1}, a_i)_c = (a_2, a_3, \ldots, a_i, a_1)_c = \ldots = (a_i, ...

**0**

votes

**1**answer

124 views

### For a set of positive integers, is this inequality always true?

The input consists of a set of positive integers $\{b_1,...,b_2\}$ such that $$\sum_{i=1}^nb_i=CK,$$ with $C$ and $K$ two positive integers.
The question is the following, is there $k\in\{1,...,n\}$ ...

**3**

votes

**1**answer

134 views

### Asymptotic for binomial sums

Let $S(n, t) = \sum_{k = 0}^n {n \choose k} ^t$.
The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.
Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.
For $S(n,2)$ we can use ...

**5**

votes

**1**answer

75 views

### Reference for restriction formula in terms of double Schubert polynomials

Everyone (that is, everyone who cares) knows that double Schubert polynomials represent Schubert classes in equivariant cohomology in type $A$. We also know that we can restrict Schubert classes to ...

**4**

votes

**1**answer

201 views

### Summation of an infinite q-series

When calculating a Partition function, I encounter the following summation
$$\sum_{n=0}^{\infty} x^n q^{n^2}.$$
I know that the sum $\sum_{n=-\infty}^{\infty} x^n q^{n^2}$ is a Theta function, but I ...

**6**

votes

**0**answers

87 views

### By replacing the Laver tables with nearly distributive fake Laver tables, can one produce algebras where the period of 1 grows fast?

I am now generalizing the notion of the classical Laver table and the fake Laver table to a larger class of algebras. A sequence of algebras $(\{1,...,2^{n}\},*_{n})_{n\in\omega}$ is said to be a ...

**6**

votes

**0**answers

115 views

### How distributive are the fake Laver tables?

The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$.
Let's now replace the Laver table $A_{n}$ with ...

**5**

votes

**1**answer

168 views

### Is this subset-sum-type problem discussed in the literature?

Let $y \in \mathbb{Z}_+^n$, with $y_1 < \dots < y_n$. I am interested in finding a 0-1 matrix $A$ and $x \in \mathbb{Z}_+^m$ s.t. $m$ is minimal and $Ax = y$, where I am guaranteed that at least ...

**3**

votes

**0**answers

120 views

### How to prove $\prod_{\lambda\vdash n}\prod_im_i(\lambda)!=\prod_{\lambda\vdash n}1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots$

Let $\lambda$ be a partition of an positive integer $n$, it can be presented as $\lambda=(\lambda_{1},\lambda_2,\cdots,\lambda_l)$ such that $\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_l>0$, or ...

**3**

votes

**0**answers

47 views

### How to prove that $\sum_{i=0}^n\frac{(a;q)_i}{(q;q)_i}\frac{(b;q)_{n-i}}{(q;q)_{n-i}}a^{n-i}=\frac{(ab;q)_n}{(q;q)_n}$?

By Cauchy identity, $${}_1\phi_0(a;—;q,z)=\sum_{n\geq0}\frac{(a;q)_n}{(q;q)_n}z^n=\frac{(az;q)_{\infty}}{(z;q)_\infty},\quad|z|<1,|q|<1,$$
we can obtain the $q-$analogue of ...

**5**

votes

**0**answers

97 views

### (Double) Crystal reflection operators on SSYTs

I am not that familiar with the language of crystals, but this is what I know:
Let $SSYT(\lambda, \mu)$ be the set of semi-standard Young tableaux with shape $\lambda$ and weight $\mu$.
There are ...

**6**

votes

**2**answers

333 views

### Sums of reciprocals of products of factorials

Let $d,m, r$ be positive integers, and define
$$
S = \left\{ (i_1, i_2, \dots, i_m) \in {\bf Z}_{+}^{m} \left | \sum_j i_j = d; \& \forall j, i_j \leq r \right. \right\};
$$
Here ${\bf Z}_+$ ...

**8**

votes

**0**answers

101 views

### Product of Partial Orders

Define the transpose product of a partial order $P$ over a set $S$ in the following way. The direct product of a partial order $P \subseteq S \times S$ and its converse, $P^{op}$, gives a partial ...

**7**

votes

**2**answers

137 views

### $f$-vector of simple convex polytope via directions of facets

Let $P$ be a simple convex polytope in $\mathbb{R}^d$ (that is, any vertex belongs to exactly $d+1$ facets). Given the collection of outer normals to facets of $P$, combinatorics of $P$ may be ...

**1**

vote

**1**answer

73 views

### Probability of paths to the boundary of a tree

Let $G_n$ be the $4$-regular tree of depth $n$, that is to say the finite graph given by the ball of radius $n$ in the Cayley graph of the free group on two generators. By the root I mean the vertex ...

**5**

votes

**1**answer

154 views

### Intersection of rotating regular polygons

This question has a recreational flavor, but may not be
entirely uninteresting.
Let $P_k$ be a unit-radius regular polygon of $k$ sides,
and $P_n$ a unit-radius regular polygon of $n \ge k$ sides.
...

**12**

votes

**2**answers

833 views

### On Hamkins' answer to a problem by Michael Hardy

Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum ...