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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9
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0answers
342 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 ...
4
votes
2answers
107 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 ...
7
votes
2answers
131 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 ...
4
votes
1answer
172 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
2answers
82 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
1answer
111 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 ...
7
votes
1answer
98 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 ...
0
votes
0answers
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
0answers
43 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 $k$ ...
3
votes
1answer
66 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 ...
1
vote
1answer
75 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 ...
18
votes
0answers
251 views

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 = ...
3
votes
0answers
168 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 ...
3
votes
2answers
127 views

The Hadwiger number of $L(K_n)$

For $n\in\mathbb{N}$ we consider the set $\{1,\ldots,n\}$ and define the line graph $L(K_n)$ of the complete graph $K_n$ as follows: $V(L(K_n)) = \big\{\{a,b\}: a,b\in \{1,\ldots, n\}, a\neq b ...
4
votes
1answer
105 views

Segments on a family of parallel lines

Let $\{l_i:i\in I\}$ be a family of parallel lines on the plane $\mathbb{R}^2$. Suppose for each $i\in I$ there is a closed segment $s_i\subset l_i$. Moreover, for each triple $i_1,i_2,i_3$ there ...
3
votes
3answers
327 views

When few simple conditions yield a unique intricate structure

If people were asked to do a brainstorming related to the title, everyone would probably come up with dozens of examples. Those could include things as different as the Mandelbrot set, Julia sets ...
1
vote
1answer
112 views

Is the number of vertices bounded for fixed max degree and fixed diameter?

Are there positive integers $\Delta, d$ such that the following statement is true? For every $n\in \mathbb{N}$ there is a graph $G = (V,E)$ such that $|V| = n$, $\Delta(G) \leq \Delta$ ...
5
votes
4answers
295 views

Is there a non-explicit characterization of the Specht modules?

It is a basic fact about the symmetric group $S_n$ that its irreducible representations are indexed by partitions of $n$. My question is, can the association between partitions and irreps be ...
4
votes
0answers
103 views

Pattern avoidance in Young diagrams

There has been lots of studies on pattern-avoiding permutations. There is also some work done on permutations whose permutation matrix fit in a diagram shape, and then do pattern avoidance on this ...
0
votes
1answer
57 views

Edge-disjoint cycles in graphs

Given a graph $G=(V,E)$ and a fixed integer $k$ are there any algorithms known which would find the maximum number of edge-disjoint cycles of length $k$ in $G$? If not is there a proof that this ...
11
votes
0answers
270 views

Transitivity of balanced mass transport in Z

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...
5
votes
1answer
106 views

Time to generate a filled-in sub-checkered board

Take an $m \times n$ checkered board and one-at-a-time add a piece to an empty square. At what point are you guaranteed to have an $s \times t$ sub-board where all of its squares are filled? Here I ...
19
votes
5answers
748 views

Three-halves-free words (analogous to square-free)

A square-free word is a string of symbols (a "word") that avoids the pattern $XX$, where $X$ is any consecutive sequence of symbols in the string. For alphabets of two symbols, the longest square-free ...
3
votes
0answers
99 views

``Occasional'' Laurent phenomenon

This question is motivated by Richard Stanley's A question on the Laurent phenomenon (motivated by his answer to the question what is the probability that a scissor became the champion?). He asked ...
0
votes
0answers
37 views

Is there any analytically expressible choice of disjoint perfect matchings?

Consider being given a $d-$regular $(n,n)$-bipartite graph. We know that its edge set decomposes into $d$ disjoint perfect matchings. I want to know if there is a analytic way to pick such a ...
4
votes
1answer
106 views

Estimating the distribution of minimal hamming distances within a set of strings?

Is their an efficient mathematical way to estimate the distribution of minimal hamming distances for a set of random strings of length 8 over a 4-letter alphabet? E.g. given a set of 100-10,000 ...
2
votes
0answers
34 views

Is the size of maximum matching in vertex transitive 3-uniform hyper-graph on $n$ vertices always $\Omega(n)$?

What is the best known lower bound on the size of the maximum matching in a vertex transitive $3$-uniform hyper-graph?
3
votes
1answer
116 views

convex decompositions of the sphere

Consider a decomposition of the sphere $S^n$ into convex pieces (that is, every cell of the cell decomposition is convex, in particular, is contained in a hemisphere). Consider the $k$-skeleta of ...
8
votes
1answer
274 views

Transitive closure of balanced mass transport in Z (move to close)

Given two atomic measures $\mu$ and $\nu$ on $\mathbb{Z}$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ along with ...
3
votes
1answer
64 views

Dense high-degree sub-graphs of dense graphs

Let $G$ be a graph with $n$ vertices and $m$ edges, and let $d=\lfloor\frac{m}{n}\rfloor$ be the rounded-down average-degree. A lemma that is attributed to Erdos says that $G$ has a non-empty induced ...
0
votes
0answers
101 views

How large can a set of nearly equidistant points be?

Suppose that $D$ is a set of points in $\mathbb{R}^{k}$ such that all pairwise distances between them belong to $[1,1+\epsilon]$. It seems that such a set cannot be very large and that its ...
3
votes
0answers
121 views

Is there any good survey on the hook length formula and related topics?

I am recently doing some research related to the hook length formula. The hook formula counts the number of Young tableaux of certain type. I find there are plenty of research already been done and ...
37
votes
1answer
3k views

Mathematicians wearing hats on arbitrary total orders

I've been pondering the following generalisation of a famous problem (the special case where $T = \mathbb{N})$: Question: We have some totally-ordered set $T$ of mathematicians, each wearing a hat ...
3
votes
0answers
55 views

Matroid rank decay

Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field ...
2
votes
0answers
56 views

Applications of small Kakeya sets over finite fields

It was proved by Dvir that a Kakeya set in $\mathbb{F}_q^n$ has size at least $q^n/n!$, a bound which was later improved to $q^n/2^n$. For $n = 2$ and $q$ odd the exact bound is $q(q+1)/2 + (q-1)/2$ ...
5
votes
2answers
158 views

Combinatorial identity and Fuss-Catalan numbers

I would like to show that $$ \lim_{N\to\infty}\frac{1}{N^{np+1}}\frac1{p!}\sum_{j=0}^{p-1}(-1)^j\binom{p-1}{j} \left(\frac{\Gamma(N+p-j)}{\Gamma(N-j)}\right)^{n+1} =\frac1{np+1}\binom{(n+1)p}{p}, $$ ...
4
votes
0answers
166 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 ...
1
vote
0answers
128 views

Generalization of Little Fermat Theorem for a particular $a$ and perfect shuffles

I'm looking for the smallest $n\in \mathbb{N}$ that solves the following equation: $$2^n=1 \mod m$$ For an odd $m$. I know that Little Fermat Theorem and Euler Totient give me a solution but they ...
1
vote
0answers
163 views

What is the status on questions related to Bhargava's factorial function?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like: For $k, l \in \mathbb{Z}$, we have $k! \times ...
1
vote
0answers
68 views

Notions of consistency / heterogeneity in sets of vector values?

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
-1
votes
0answers
23 views

Deep questions in number theory not accessible by combinatorial results

Number theory and arithmetic geometry were invented to solve many questions about properties of numbers. What are the some of the foundational results or estimates that are accessible to powerful ...
1
vote
1answer
99 views

Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
7
votes
0answers
126 views

Combinatorics of palindromic decompositions

This is sort of a companion to my question Number of trivializations of a trivial word in the free group (which in turn is motivated by my earlier question here). It turns out that that question may ...
16
votes
7answers
1k views

Are there any Algebraic Geometry Theorems that were proved using Combinatorics?

I'm collaborating with some algebraic geometers in a paper, and when writing the introduction I mentioned the interaction of Combinatorics and Algebraic Geometry, and gave some examples like the ...
-2
votes
1answer
108 views

About structure of the set of perfect matchings of $K_{n,n}$

Are there any special properties known about the set of perfect matchings of $K_{n,n}$? Like any global structure of this set? Some natural way to partition it? Like is there some algebraic structure ...
6
votes
0answers
225 views

Mass Transportation Through Wonderful Roller

There is a wonderful roller that transports mass from A to B and there is a pile of sand with weight $W$ in point A and we want to transport it to B. Wonderfulness of roller comes from this property ...
0
votes
1answer
67 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 $f:\mathcal{F}\rightarrow \Omega$, ...
2
votes
1answer
61 views

Is the union of strongly base-orderable matroids strongly base-orderable?

A matroid is said to be strongly base-orderable if for any two bases $B_1,B_2$ there is a bijection $f:B_1 \to B_2$ such that for any $S\subseteq B_1$ set $(B_1 \setminus S) \cup f(S)$ is also a base. ...
5
votes
0answers
96 views

For which sidelengths are there polyominos composed of three squares that tile the plane?

Given three naturals $a<b<c$. We consider polyominos, connected or not, which are composed of three squares of sides $a,b,c$. How can one characterize all triples $a,b,c$ for which such a ...
-1
votes
1answer
114 views

How compute combinatorial expression [closed]

How compute $\sum_{j=1}^k \binom{x}{j}\binom{k-1}{j-1}\alpha^j, \quad x, \alpha\in\mathbb{R}$