# Questions tagged [co.combinatorics]

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

1,666
questions with no upvoted or accepted answers

**56**

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### Which region in the plane with a given area has the most domino tilings?

I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to ...

**50**

votes

**1**answer

3k views

### (Approximately) bijective proof of $\zeta(2)=\pi^2/6$ ?

Given $A,B\in {\Bbb Z}^2$, write $A \leftrightarrows B$ if the
interior of the line segment AB misses
${\Bbb Z}^2$.
For $r>0$, define
$S_r:=\{ \{A, B\} | A,B\in {\Bbb Z}^2,||A||<r,||B||<r, |...

**44**

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1k views

### Does every triangle-free graph with maximum degree at most 6 have a 5-colouring?

A very specific case of Reed's Conjecture
Reed's $\omega$,$\Delta$, $\chi$ conjecture proposes that every graph has $\chi \leq \lceil \tfrac 12(\Delta+1+\omega)\rceil$. Here $\chi$ is the chromatic ...

**34**

votes

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810 views

### Is there anything to the obvious analogy between Joyal's combinatorial species and Goodwillie calculus?

Combinatorial species and calculus of functors both take the viewpoint that many interesting functors can be expanded in a kind of Taylor series. Many operations familiar from actual calculus can be ...

**31**

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1k 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 = (\...

**30**

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**0**answers

2k views

### A Combinatorial Abstraction for The “Polynomial Hirsch Conjecture”

Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ .
Suppose that
(*)
For every $i \lt j \lt k$
and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, ...

**26**

votes

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941 views

### How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?

For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have
$$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$
Note that this ...

**26**

votes

**0**answers

744 views

### A question on simultaneous conjugation of permutations

Given $a,b\in S_n$ such that their commutator has at least $n-4$ fixed points, is there an element $z\in S_n$ such that $a^z=a^{-1}$, and $b^z=b^{-1}$? Here $a^z:=z^{-1}az$.
Magma says that the ...

**24**

votes

**0**answers

628 views

### Minimal number of intersections in a convex $n$-gon?

For a convex polygon $P$, draw all the diagonals of $P$ and consider the intersection points made by those diagonals. Let $f(n)$ be the minimal number of such intersections where $P$ ranges over all ...

**24**

votes

**0**answers

1k views

### Do all possible trees arise as orbit trees of some permutation groups?

I.Motivation from descriptive set theory
(Contains some quotes from Maciej Malicki's paper.)
The classical theorem of Birkhoff-Kakutani implies that every metrizable topological group G admits a ...

**23**

votes

**0**answers

899 views

### Linking formulas by Euler, Pólya, Nekrasov-Okounkov

Consider the formal product
$$F(t,x,z):=\prod_{j=0}^{\infty}(1-tx^j)^{z-1}.$$
(a) If $z=2$ then on the one hand we get Euler's
$$F(t,x,2)=\sum_{n\geq0}\frac{(-1)^nx^{\binom{n}2}}{(x;x)_n}t^n,$$
on the ...

**23**

votes

**0**answers

697 views

### Which sets of roots of unity give a polynomial with nonnegative coefficients?

The question in brief: When does a subset $S$ of the complex $n$th roots of unity have the property that
$$\prod_{\alpha\, \in \,S} (z-\alpha)$$
gives a polynomial in $\mathbb R[z]$ with ...

**23**

votes

**0**answers

2k views

### Why do polytopes pop up in Lagrange inversion?

I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...

**22**

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**0**answers

571 views

### How much of the plane is 4-colorable?

In 1981, Falconer proved that the measurable chromatic number of the plane is at least 5. That is, there are no measurable sets $A_1,A_2,A_3,A_4\subseteq\mathbb{R}^2$, each avoiding unit distances, ...

**22**

votes

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436 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 ...

**22**

votes

**0**answers

712 views

### Combinatorics of Quantum Schubert Polynomials

Let $S_n$ be the symmetric group. Let $s_i$ denote the adjacent transposition $(i \ i+1)$. For any permutation $w\in S_n$, an expression $w=s_{i_1}s_{i_2}\cdots s_{i_p}$ of minimal possible length is ...

**22**

votes

**0**answers

2k views

### The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...

**21**

votes

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412 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 ...

**21**

votes

**0**answers

1k views

### What is the best lower bound for 3-sunflowers?

A collection of $t$ sets $A_i$ is called a t-sunflower if $A_i \cap A_j = Z $ for all $i \neq j$ for some fixed $Z$. A well-known conjecture of Erdos and Rado says that there is a constant $C_t$ such ...

**21**

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400 views

### Is the Poset of Graphs Automorphism-free?

For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$.
Is ...

**21**

votes

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473 views

### Hall's Marriage Theorem and intervals

In Hall's Marriage Theorem, we have a set $B$ of brides and $G$ of
grooms, where each bride $b$ has an acceptable set $A_b \subseteq G$
of grooms. A matching $m:B\to G$ is an injection such that $m(b)
...

**21**

votes

**0**answers

515 views

### A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...

**20**

votes

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782 views

### Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$
Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by
\begin{equation*}
e_k(\vx) := \sum_{1 \...

**20**

votes

**0**answers

2k views

### Trigonometry related to Rogers--Ramanujan identities

For integers $n\ge2$ and $k\ge2$, fix the notation
$$
[m]=\sin\frac{\pi m}{nk+1} \quad\text{and}\quad
[m]!=[1][2]\dots[m], \qquad m\in\mathbb Z_{>0}.
$$
Consider the following trigonometric numbers:...

**19**

votes

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301 views

### “quantum” symmetric plane partitions beget alternating sign matrices?

The "quantum" version qTSPP of the number of totally symmetric plane partitions, contained in the cube $[0,n]^3$, is enumerated by
$$f_n(q):=\prod_{j=1}^n\prod_{k=1}^j\prod_{\ell=1}^k\frac{1-q^{j+k+\...

**19**

votes

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680 views

### A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions
$$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$
Suppose we say that $...

**19**

votes

**0**answers

2k views

### Origins of the Nerve Theorem

Recently, I've read two papers which have cited the Nerve Theorem, one crediting Borsuk with the result and another Leray. Here is the question:
Who was the first to prove the Nerve Theorem?

**18**

votes

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406 views

### Simpler proofs of certain Ramsey numbers

The reason for the gorgeous simplicity of the classic proofs of $R(3,3)$, $R(4,4)$, $R(3,4)$ and $R(3,5)$ is that essentially all you need is the trivial bound and a picture.
But for bigger Ramsey ...

**18**

votes

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372 views

### An algebraic strengthening of the Saturation Conjecture

The Saturation Conjecture (proved by Knutson-Tao) asserts that
$c_{n\mu,n\nu}^{n\lambda}\neq 0\Rightarrow c_{\mu,\nu}^{\lambda} \neq
0$, where $c$ denotes a Littlewood-Richardson coefficient and $n$ ...

**18**

votes

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314 views

### Deforming a basis of a polynomial ring

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions $\{S_\...

**17**

votes

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523 views

### Ramsey's theorem for the first uncountable ordinal

Sierpiński proved that a version of Ramsey's theorem for colourings of pairs of countable ordinals fails miserably by comparing the ordering of $\omega_1$ with the linear ordering of (a subset of) the ...

**17**

votes

**0**answers

466 views

### Question about combinatorics on words

Let $\{a_1,a_2,...,a_n\}$ be an alphabet and let $\{u_1,...,u_n\}$ be words in this alphabet, and $a_i\mapsto u_i$ be a substitution $\phi$.
Question: Is there an algorithm to check if for some $m,k$...

**17**

votes

**0**answers

410 views

### Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...

**17**

votes

**0**answers

608 views

### Probability of many overlapping zero inner products on a circle

[Question edited and changed a little on June 14 2015]
Consider an $n$-dimensional vector $v$ with $v_i \in \{-1,1\}$. Now consider an $n$-dimensional vector $w$ with $w_i \in \{-1,0,1\}$. The ...

**17**

votes

**0**answers

732 views

### What's the big deal about $M_{13}$?

$M_{13}$ is the Mathieu groupoid defined by Conway in
Conway, J. H. $M_{13}$. Surveys in combinatorics, 1997 (London), 1–11,
London Math. Soc. Lecture Note Ser., 241, Cambridge Univ. Press, ...

**17**

votes

**0**answers

660 views

### Is there a n/2 version of the Erdős-Hanani conjecture?

This question comes out of REU research from this past summer. Unfortunately weeks of thought led to only trivial observations and the conclusion that the problem is quite hard.
Fix $k,t$. Let $F$ be ...

**17**

votes

**0**answers

1k views

### coloring ${\mathbb Z}^k$

This question is related to but seems to be simpler than this one, so perhaps somebody can solve it.
Question. Is there $k\ge 1$ and a coloring of vertices of the lattice ${\mathbb Z}^k$ in $k$ ...

**17**

votes

**0**answers

746 views

### Combinatorial identity involving the Coxeter numbers of root systems

The setup is:
$R$ = irreducible (reduced) root system;
$D$ = connected Dynkin diagram of $R$, with nodes numbered $1,2,...,r$;
$\hat D$ = extended Dynkin diagram, nodes numbered $0,1,2,...,r$;
$\...

**16**

votes

**0**answers

264 views

### Straight-line drawing of regular polyhedra

Find the minimum number of straight lines needed to cover a crossing-free straight-line drawing of the icosahedron $(13\dots 15)$ and of the dodecahedron $(9\dots 10)$ (in the plane).
For example, ...

**16**

votes

**0**answers

861 views

### What to do with results you found but cannot prove(outside your research area)?

Not sure if MathOverflow is still a place to discuss such things, but I'll give it a try. Tell me an alternative site, in case it is wrong here. I translated a representation-theory/combinatorial ...

**16**

votes

**0**answers

401 views

### A Product Related to Unrestricted Partitions

Start with the product for unrestricted partitions:
$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...

**16**

votes

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398 views

### Maximum automorphism group for a 3-connected cubic graph

The following arose as a side issue in a project on graph reconstruction.
Problem: Let $a(n)$ be the greatest order of the automorphism group of a 3-connected cubic graph with $n$ vertices. Find a ...

**16**

votes

**0**answers

694 views

### How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highest power of 2 which ...

**15**

votes

**1**answer

510 views

### To find a longer path with fixed endvertices in a graph satisfies the following property

Suppose that $G=(V,E)$ is a simple graph and $P=(V_1,E_1)$ is a path in $G$ where
$$V_1=\{v_0,v_1,\cdots,v_n\},\ E_1=\{v_0v_1,v_1v_2,\cdots,v_{n-1}v_n\}.$$
I found that if the path $P$ satisfies:
...

**15**

votes

**0**answers

292 views

### Combinatorial characterization of intersecting intervals in the plane

Consider $n$ points $A=\{A_1,\dotsc,A_n\}$, and another set of points, $B=\{B_1,\dotsc,B_n\}$ in the plane. We can assume they are all disjoint.
For each permutation $\pi$, consider the collection of ...

**15**

votes

**0**answers

421 views

### Word complexity of primes mod 4

For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...

**15**

votes

**0**answers

226 views

### realization spaces of 3-dimensional polytopes

It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible.
A proof of this theorem can be found for instance in ...

**15**

votes

**0**answers

501 views

### Catalan objects associated to a univariate polynomial

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data:
a noncrossing matching on $2n$ ...

**15**

votes

**0**answers

2k views

### Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...

**15**

votes

**0**answers

864 views

### Polynomials with presumably positive coefficients

After seeing that some positivity problems get their solutions on MO,
I am quite enthusiastic of posing my (and not only) problem of positive flavour.
In order to state it, I have to introduce the ...