# Tagged Questions

**7**

votes

**0**answers

33 views

### Mysterious identity between numbers of odd/even meander systems

Definitions:
An upper arch system of order $n$ is a subset of the plane consisting of $n$ non-intersecting semicircles in the upper half-plane whose endpoints belong to the set $\{(k,0)\mid ...

**11**

votes

**2**answers

300 views

### A combinatorial question about orthonormal bases

Suppose that $F:S^{n-1}\to A$ is a map of sets from the unit sphere in $\mathbb R^n$ to an abelian group, and that the sum $F(v_1)+\dots +F(v_n)$ over an orthonormal basis is independent of the basis. ...

**3**

votes

**0**answers

32 views

### Connection between connectivity and cohesion of a graph

Tutte [1] proved that, for every $3$-connected graph $G$ and vertices $u$ and $v$, there exists a nonseparating $uv$-path.
A graph $G$ is $t$-cohesive if $G$ is connected, has at least two vertices, ...

**6**

votes

**0**answers

75 views

### Coloring a Ferrers diagram

I've shopped the problem below around a bit and it seems like it might be known, or not that hard to resolve, but so far I've come up empty-handed.
Say that a coloring of the dots of a Ferrers ...

**18**

votes

**1**answer

874 views

### A Question on 1, 2 ,3 Conjecture

The 1, 2, 3 conjecture is well-known:
If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the ...

**4**

votes

**1**answer

271 views

### The number of monotone full binary trees

Let $\rho$ be an equivalence relation on a semigroup $S$. A subsemigroup $S'$ of $S$ is called a $\rho$-cross-section of $S$, provided that $S'$ contains exactly one representative from each ...

**1**

vote

**1**answer

103 views

### Existence of functions on finite sets with specific propertise

Let $\Omega$ be an universal set and $|\Omega|=N$, denote $\mathcal{F}$ to be the family of all subsets $\subset \Omega$ with cardinal $n$. We now define a function $f:\mathcal{F}\rightarrow \Omega$, ...

**3**

votes

**1**answer

112 views

### A question of terminology regarding integer partitions

I am wondering if there is a standard notation and name for the following. Let $\lambda$ be a partition $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_r\geq 1$ of $n$ into $r$ parts. Then we can ...

**8**

votes

**1**answer

283 views

### Number of trivializations of a trivial word in the free group

Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...

**4**

votes

**2**answers

146 views

### Ordered lattice point enumeration

I initially asked this question over at StackOverflow as it has algorithmic flavor to it, but I haven't been getting much traction so I thought I would probe the mathematics community.
Setup: Let ...

**4**

votes

**1**answer

285 views

### Generating non-isomorphic graphs by adding edges to a given graph

This question is in a way related to the one I posted on math.se. Since the question there did not produce any final answer I am trying my luck here!
I am given a fairly large graph $G$ and subsets ...

**3**

votes

**1**answer

104 views

### Regarding left-to-right minima

Let $\rho$ be a permutation on $[1,n]$ and $l_i$ be the number of left-to-right minima in $\rho_{i\ldots n}$, I know that for a random permutation $E[l_1] = H_n$ (the $n$-th Harmonic number) but is ...

**18**

votes

**2**answers

1k views

### An unfair marriage lemma

I am looking for a citeable reference to the following generalization of Hall's Marriage Theorem:
Given a bipartite graph of boys and girls. In addition to gender difference, they are divided into ...

**11**

votes

**2**answers

580 views

### Gale-Shapley stable marriage theorem: can we entrust matchmaking to monkeys?

Disclaimer: This is a question I have not done any real research about. I asked it myself some 5 years ago, and back then I had no idea where to start. Now I have some texts on stable matchings lying ...

**3**

votes

**1**answer

382 views

### Simplest form for sum of Binomial Expressions

How difficult is the problem of reducing the number of terms in a sum of binomial expressions? Formally:
Given $a_1, a_2, a_3, … a_n$, and $b_1, b_2, b_3, ... , b_n$, where $a_i, b_i \in \mathbb{Z}$, ...

**7**

votes

**0**answers

99 views

### Some $q-$analogues of $ \sum\limits_{j = - k}^k {{{( - 1)}^{ j}}}\binom{n}{k-j}\binom{n}{k+j}=\binom{n}{k}.$

Let ${\left( {a;q} \right)_n}=\prod\limits_{j = 0}^{n - 1} {(1-{q^j}a} )$ and
let $ {{n}\brack{k}}_q$ denote a $q-$binomial coefficient.
I am interested in $q-$analogues of the identity $ ...

**1**

vote

**1**answer

70 views

### Balls from bin with replacement, distinct elements, concentration inequality

Draw $n$ numbers, denoted by $a_1, a_2, \ldots, a_n$, from set $[n]$, that is, for each $i$, $a_i$ is a uniformly random number from $[n]$.
Let $A = \{a_1, a_2, \ldots, a_n\}$. Then
$$
...

**15**

votes

**1**answer

814 views

### Evaluating a remarkable term for primes p = 5 (mod. 8)

Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of
$$w_p = \prod_{a \in (\mathbb F_p^{\times})^2}(1 + \zeta^a) + \prod_{b ...

**2**

votes

**0**answers

54 views

### Edge-disjoint path-systems in infinite digraphs

Let $ D=(V,A) $ be a directed graph without backward-infinite paths and let $ \{ s_i \}_{i<\lambda},U \subset V $ where $
\lambda $ is some cardinal. Assume that for all $ u\in U $ there is a ...

**11**

votes

**2**answers

441 views

### “The Two Sheriffs” puzzle -2: threshold for security

I've already asked a question “The Two Sheriffs” puzzle with wrong assumption. Yoav Kallus in his amazing answer using Fano plane showed that the problem has a solution in the case of seven suspects.
...

**3**

votes

**0**answers

82 views

### What is known about the complexity of this covering problem?

Let $G=(V,E)$ be a graph. A vertex set $X\subseteq V$ is called critical if $X\neq\emptyset$ and no vertex in $V\setminus X$ is adjacent to exactly one vertex in $X$. The problem is to find a vertex ...

**7**

votes

**1**answer

137 views

### Determinant of a checkerboard Hankel matrix with Catalan numbers

My goal is to compute
\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
...

**5**

votes

**0**answers

80 views

### The Maximum Number of Lines Contained in the Point Set of a Finite Projective Plane

Consider a finite projective plane of order $q$. Define $f(m)$ to be the maximum number of lines completely contained in any point set of size $m$, where $1 \leq m \leq q^2+q+1$. I would like to ...

**24**

votes

**0**answers

497 views

+250

### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

**0**

votes

**1**answer

207 views

### Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...

**0**

votes

**1**answer

66 views

### Counting the orderings of outward-directed trees where the degree of each vertex is $2$

Let $T$ be a connected directed tree with the following properties:
The degree of each vertex of $T$ is at most $2$ (I am sure there is a name for such a graph but I do not know it).
$T$ has a ...

**3**

votes

**1**answer

135 views

### Enumerating matrices function of ranks

Is there an expression/approximate expression for number of real matrices $M\in\{0,1\}^{n\times n}$ of rank $r\leq n$?
Is it known if $M$ is restricted to symmetric/skew-symmetric matrices?
Does ...

**0**

votes

**2**answers

99 views

### Combinatorial Interpretation of Generalized Stirling numbers

I know the combinatorial interpretation of first, and second order Stirling numbers (#of k cycles of n items, and #of partitions n items into k subsets). Is there an interpretation for the generalized ...

**3**

votes

**0**answers

117 views

### A Result of Anders Bjorner: Matchings in countably infinite geometric lattices of finite height

Let $L$ be a countably infinite geometric lattice of finite height $r\ge3$. (A geometric lattice of height $r$ is an atomistic semimodular lattice such that every maximal chain has $r+1$ elements.)
...

**0**

votes

**1**answer

280 views

### Maximize combinatorial sum for boolean function

I am trying to maximize the function
$$ S(f)=\sum_{j=0}^{n-\frac{n-1}{t}}(-1)^j{n-\frac{n-1}{t}\choose{j}}\sum_{i=0}^{\frac{n-1}{t}}(-1)^{f(i-j)}(t-1)^i{\frac{n-1}{t}\choose{i}} $$
for a function ...

**26**

votes

**1**answer

1k views

### “The Two Sheriffs” puzzle

This puzzle is taken from the book Mathematical puzzles: a connoisseur's collection by P. Winkler.
Two sheriffs in neighboring towns are on the track of a killer, in a
case involving eight ...

**5**

votes

**0**answers

244 views

### Have topographs been studied before?

This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...

**7**

votes

**2**answers

155 views

### Decomposition of a cross-polytope into simplices

Consider the standard embedded $n$-cross polytope $P_n$ with vertices $\pm e_i \in \mathbb R^n$. Let us consider decompositions of this polytope into $2^{n-1}$ simplices, such that these simplices ...

**7**

votes

**1**answer

112 views

### Is there a Degenerate Dependency Local Lemma?

The Lovasz Local Lemma has several generalizations, with names usually starting with L, such as Lopsided or Lefthanded.
Here I ask whether another possible generalization (for which I could not yet ...

**1**

vote

**1**answer

106 views

### Hamming graph and independent sets

I'm defining the Hamming graph $H(d,q)$ in the usual way, so we have a set $S$ of $q$ elements, the hamming graph $H(d,q)$ has vertex set $S^{d}$ (the set of all ordered $d$-tuples of elements of $S$) ...

**3**

votes

**0**answers

191 views

### adding a boundary to the finite upper half-plane

Let $\Bbb{F}_q$ be a finite field, let $\delta \in \Bbb{F}_q$ be a non-square, let $\Bbb{F}_{q^2} = \Bbb{F}_q\big( \sqrt{\delta} \big)$ be the corresponding quadratic extension,
and let ...

**4**

votes

**2**answers

135 views

### How many simple cycles can a graph with $n$ vertices and $m$ edges have?

I am mainly interested in the smallest number of simple cycles a graph with $n$ vertices and $m$ edges must have.
For example, if $m\le n-1$, this number is $0$, then if $n\le m \le 3(n-1)/2$, it is ...

**24**

votes

**3**answers

490 views

### Ordering subsets of the cyclic group to give distinct partial sums

Suppose that you are given a set $S$ of $k$ nonzero elements from $\mathbb{Z}_n$. Is it always possible to order the elements of $S$, say $a_1,a_2,\dots,a_k$ in such a way that the partial sums ...

**1**

vote

**1**answer

92 views

### Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory

Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let ...

**4**

votes

**1**answer

189 views

### Young tableau with no i in row i, name that derangement

This question has its genesis in a group assignment: $k$ students are to be given oral exams. Each student will be asked one distinct question from $n$ questions given to them earlier, no two students ...

**6**

votes

**2**answers

99 views

### Reference request: monochromatic paths in edge-colored complete graphs

Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$.
...

**3**

votes

**1**answer

72 views

### congruence for Stirling numbers of the first kind

Suppose we write $s(n,k)$ for the Stirling numbers of the first kind. When $p$ is a prime number, I'm interested in knowing when $s(p^a, k)$ is divisible by $p^a$. So:
What is known about ...

**3**

votes

**1**answer

128 views

### Identify ring of polynomials symmetric under forgetting variables

I came across the following ring $A$, which appears as a Chow ring. I am wondering if it has been studied before; in particular, I am looking for a reference where this object might have been ...

**2**

votes

**2**answers

571 views

### inverted factorial and trailing zeros problem

First than anything a big Hello for all math fans like me.
I've found a problem that is pretty interesting and I can't find the answer.
As all of you must know, to counting the trailing zeros of $n$ ...

**2**

votes

**3**answers

375 views

### Characterizing Posets by Functions Into Natural Numbers

Let $P$ be a poset and denote by $Hom(P, \mathbb N)$ the set of all monotone functions from $P$ to natural numbers $\mathbb N$. Under what conditions on $P$ Is it possible to recover the order on $P$ ...

**0**

votes

**0**answers

31 views

### Number of ways you can form pairs with a group of people when certain people cannot be paired with each other [migrated]

Let's say you have a group of eight people and you want to form them into pairs for group projects.
There are
8!/(4!*2!)
ways to do it. (8! Is the total ...

**2**

votes

**0**answers

47 views

### Bounds on number of distinct substrings

I have a table with $r$ rows of length $\ell$, with each cell containing a letter from an alphabet $A$ of length $a$. I'm trying to determine the expected number of distinct strings of length $k$ ...

**5**

votes

**6**answers

669 views

### Faa di Bruno's formula for inverse functions ?

It is easy to get a expression for the nth-derivative of an inverse fuction ; starting from $(f^{-1})'=\frac{1}{f'\circ f^{-1}}$, one gets things like $(f^{-1})^{(n)}=\frac{\sum a_k\prod ...

**17**

votes

**4**answers

1k views

### What is the minimum N for which there exist N points in the plane that cannot be covered by any number of non-overlapping closed unit discs?

This problem was posed in March 2010 at G4G9 in a talk by the Japanese mathematician Hirokazu "Iwahiro" Iwasawa. He claims there is a simple proof that N > 10, though he did not share it with the ...

**1**

vote

**1**answer

155 views

### Regular unimodular triangulation for a certain simplex

Consider an $n$-simplex with vertices given by
$(0,0,\dots,r_i,r_{i+1},\dots,r_n)$ where $r_1,\dots,r_n$ are given natural numbers,
and $i=0,1,\dots,n+1$.
Does this simplex admit a regular, ...