# Tagged Questions

**11**

votes

**0**answers

301 views

### Does Langton's ant cover every n by 6 gridded torus?

This post follows this other post about times cover by Langton's ant of $n$ by $n$ gridded torus.
For $n$ by $n$ gridded torus, I've checked for $n \le 1000$ that the ant covers all. This fact needs ...

**8**

votes

**0**answers

155 views

### What's the analogue of a Young symmetrizer in the Brauer algebra?

According to Schur--Weyl duality, the centralizer of $\mathrm{GL}(V)$ acting diagonally on $V^{\otimes N}$ is the group algebra of the symmetric group $\mathbb S_N$. An equivalent formulation is the ...

**1**

vote

**0**answers

131 views

### Kripke frames as classes of partitions

Here's something I've been playing with off and on for a bit; I'm curious if anyone has seen it before.
For this question, a Kripke frame $K$ is a finite reflexive directed graph. (Reflexivity isn't ...

**5**

votes

**1**answer

159 views

### The Maximal $\ell_2$ norm of a signed sum of vectors

Suppose we have $n$ vectors in $\mathbb{R}^n.$ Consider the signed sum of these vectors:
$$U(s_1,\ldots,s_n)=s_1 v_1+s_2 v_2 + \ldots + s_n v_n$$
where $s_j$'s can only take values of $+1$ or $-1.$ I ...

**1**

vote

**1**answer

137 views

### How many finite subsets in $\mathbb{Z}^d$ have a given sum of squares?

Let $|\cdot|$ denote the usual norm in $\mathbb{Z}^d$. Given a finite subset $S \subset \mathbb{Z}^d$, let $\varphi(S) = \sum_{z \in S}|z|^2$. Given $m \in \mathbb{N}$, what is the size of ...

**3**

votes

**0**answers

83 views

### Looking for a natural definition of certain polynomials associated with skew Young diagrams

Consider a connected skew Young diagram in the English notation and then rotate it counterclockwise by $\pi/4$. This rotation can be avoided by simply replacing "rows" by "diagonals" in the below, but ...

**0**

votes

**0**answers

137 views

### Generating function for reciprocals of Stirling numbers?

Is there an (ordinary or exponential) generating function for the reciprocals $\frac{1}{s(n,k)}$ of the Stirling numbers of the first kind?
Also, is there some general way to find generating ...

**3**

votes

**0**answers

63 views

### Which (polytopal) fans/polytopes are secondary?

Let $P$ be a (d-1)-dimensional polytope with $n$ vertices that sits in an affine hyperplane in $\mathbb{R}^d$.
The secondary fan of $P$ is a polytopal fan in $\mathbb{R}^n$ with $d$-dimensional ...

**0**

votes

**0**answers

86 views

### Adjacency graph of a polyomino

Given a polyomino, the "adjacency graph" has one vertex for each tile and an edge connecting tiles which are adjacent (diagonal doesn't count). Is anything known about which graphs can be the ...

**19**

votes

**0**answers

444 views

### Time for Langton's ant to cover a “square” torus

Langton's ant is a cellular automaton running as follows:
Squares on a plane are colored variously either black or white. We
arbitrarily identify one square as the "ant". The ant can travel ...

**16**

votes

**0**answers

234 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 ...

**9**

votes

**5**answers

691 views

### Combinatorial\Probabilistic Proof of Stirling's Approximation

Stirling's approximation is the following well-known asymptotic result:
$$n! \approx \left(\frac{n}{e}\right)^n \sqrt{2 \pi n}$$
This result has several analytical proofs, for example via Laplace's ...

**14**

votes

**1**answer

673 views

### In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?

This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = ...

**8**

votes

**1**answer

218 views

### Yet another Erdős–Szekeres game

Given $n$. Two players in turn write different real numbers $x_1,x_2,x_3,\dots$
The player after whose turn there is a monotone subsequence of length $n$ loses.
I guess that the question 'who wins' ...

**8**

votes

**1**answer

472 views

### Erdős-Szekeres game

Given $n$. Two players in turn mark points on the plane. No three may be collinear, no $n$ may form a convex $n$-gon. The player who does not have legal move loses. Who has a winning strategy?

**11**

votes

**1**answer

471 views

### Continuous functions and 2-bushy trees

The following problem was asked by Joe Miller in the fall of 2010 at a bar in Madison.
A subtree $T \subseteq 4^{< \omega}$ is $2$-bushy if for some node $\sigma \in T$, every node above $\sigma$ ...

**0**

votes

**1**answer

91 views

### A particular argument in the review on expanders by Hoory-Linial-Wigderson

I am thinking about the third bullet point on page 455 here, http://www.ams.org/journals/bull/2006-43-04/S0273-0979-06-01126-8/
Can someone explain what is the argument there which seems to conclude ...

**4**

votes

**1**answer

489 views

### How to prove this identity on double summation series?

I suspect the following identity is valid, but I can not prove it. I just calculate it numerically.
...

**1**

vote

**1**answer

178 views

### Asymptotic of a certain double sum involving binomial coefficients

Consider sums of the form
$S(n)=\sum^{n}_{m=0}\sum^{m}_{k=1}2^{2k+m+1}{n-m+k+1 \choose 2k+2}{m \choose k}$
I am interested in the asymptotics of $S(n)$ as $n\to \infty$.
More precisely I would ...

**7**

votes

**1**answer

186 views

### Probability of a graph procedure

We are going to build $K_n$ one edge at a time. Begin with the empty graph on $n$ vertices. Take a random permutation of the edges of $K_n$ and, one at a time, place the edges onto the graph (so, ...

**3**

votes

**0**answers

110 views

### Hypercube edge-coloring problem

Question: Is there a pairing (a fixed point free involution) of the vertices of the $n$-dimensional cube graph, and a $2$-coloring of its edges such that the number of color changes needed to get from ...

**5**

votes

**1**answer

292 views

### Infinite sequence avoiding a countable set of words

As an application in group theory, I would need an infinite sequence over a finite alphabet, that avoids a sequence of words $w_i$, where the length of $w_i$ is such that $l(w_i) > 10^8 ...

**3**

votes

**3**answers

327 views

### Aperiodic graphs

The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph ...

**6**

votes

**0**answers

153 views

### Asymptotics of the multipartition function

Recall that the multipartition function $p_k(n)$ counts the number of $k$-tuples of partitions $\lambda^1,\ldots,\lambda^k$ of numbers $a_1,\ldots,a_k$ with $a_1+\cdots+a_k=n$. It has a generating ...

**1**

vote

**0**answers

31 views

### Constructing pointwise-or independent binary matrices

Let $M$ be an $m \times n$ matrix such that every $M_{ij} \in \{0,1\}$.
We say that $M$ is `$\lor$-irreducible' if (i) no row is a pointwise-or of other rows, and (ii) no column is a pointwise-or of ...

**1**

vote

**0**answers

132 views

### Experimentation with partial Euler products

Richard Mathar $[1]\& [2]$ shows that
\begin{align}
&\zeta_{2}(s)\equiv\prod_{\Omega(n)=2}^{}\left(\dfrac{1}{1 - n^{s}}\right)^{-1}=\exp \left(\sum _{k=1}^n \frac{P(k s)^2+P(2 k s)}{2 ...

**11**

votes

**0**answers

235 views

### Maximizing the number of semistandard Young tableaux

Is anything known about the following question? Given a positive
integer $p$ and a real number $0<\alpha<1$, what partition $\lambda$
whose parts sum to $\alpha p^2$ (asymptotically) and whose ...

**3**

votes

**1**answer

290 views

### Balancing real numbers in one dimension

Given numbers $0 \leq d_i \leq 1$ for $i=1,\ldots,m$, it is easy to see that you can always find signs $\varepsilon_i \in \{-1,1\}$ such that the partial sums $\sum_{i=1}^k \varepsilon_i d_i/2$, for ...

**5**

votes

**0**answers

161 views

### Staircase Schur functions squared

Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by ...

**2**

votes

**1**answer

93 views

### Relationship of clique, independence, and chromatic numbers

For any graph $G=(V,E)$ let $\bar{G}$ be the complement graph. Is $$\text{inf}\big\{\frac{\omega(G)+\omega(\bar{G})}{\chi(G)} : G \text{ is a finite graph}\big\}$$ known? If not, what lower bounds are ...

**1**

vote

**0**answers

49 views

### Select n vectors from k vectors (in 3D) such that each component of the resultant vector >= each component of a given vector M

this was left unanswered for 1 week on MStackExchange, so I thought MOverflow would be more appropriate. Thanks :)
Let $R = (R_x, R_y, R_z)$ be the resultant vector of the n vectors and $M = (M_x, ...

**5**

votes

**3**answers

295 views

### A balls and urns model for a hashing problem

Fix $N \in \mathbb{N}$. Suppose we throw $N$ numbered balls into $N$ numbered urns, so that for each $b \in \{1,\ldots,N\}$, ball $b$ lands in urn $j$ with equal probability $1/N$. Choose a number $c ...

**4**

votes

**1**answer

219 views

### Decomposing $(\mathbb C^n)^{\otimes m}$ as a representation of $S_n\times S_m$

$V=\mathbb C^n$ is a $\mathbb CS_n$-module, where $S_n$ is the symmetric group of degree $n$, via the representation sending a permutation to the corresponding permutation matrix. The tensor power ...

**5**

votes

**0**answers

116 views

### How do you categorify the cycle index series?

Let $F$ be a combinatorial species. The exponential generating series of $F$ is defined to be
$$ \sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!} $$
It was observed by Baez and Dolan in their paper ...

**-2**

votes

**1**answer

100 views

### Combinatorical meaning of such expression [closed]

Any combinatorical meaning or interpretation of
$$1^{\alpha_1}2^{\alpha_2}3^{\alpha_3}...s^{\alpha_s}\alpha_1!\alpha_2!...\alpha_s!$$
for partition ...

**6**

votes

**1**answer

265 views

### Flooding a cycle digraph via chip-firing: $n^{k-1} + n^{k-2} + \cdots + 1$ bound (a Norway 1998-99 problem generalized)

Let $k > 1$ and $n$ be positive integers. Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Let $D$ be a digraph which has exactly $k$ vertices $v_0$, $v_1$, ..., $v_{k-1}$ and exactly $k$ arcs $v_0 ...

**1**

vote

**1**answer

70 views

### Hadwiger-Nelson problem in higher dimensions

Given a positive integer $n\in \mathbb{N}$ we define the Hadwiger-Nelson graph $\text{HN}_n$ by
$V(\text{HN}_n) = \mathbb{R}^n$;
$E(\text{HN}_n) = \{\{v_1, v_2\}: v_1, v_2 \in \mathbb{R}^n \text{ ...

**5**

votes

**0**answers

110 views

### Intersections of the B-orbits and the orbits of some other Borel subgroups in the flag variety G/B

This is a follow-up of this previous question below:
Intersections of $B$ and $B^-$ orbits in the flag variety $G/B$
Let $G = SL_n(\mathbb{C})$, $B$ be the standard Borel subgroup, and consider some ...

**2**

votes

**1**answer

130 views

### Existence of Steiner system designs given $n,k,t$

I am familiar with the recent Keevash paper here which proves that given some $t,n,k,\lambda$ then provided standard divisibility conditions hold, and $n$ is suitably large, there exists a ...

**5**

votes

**2**answers

288 views

### Combinatorial designs textbook recommendation

Good evening, I am currently taking a class which has combinatorial designs as the first topic, we are using Peter Cameron's book Designs, Graphs, Codes and their Links which I am finding extremely ...

**2**

votes

**0**answers

89 views

### Adding a row to a Young Tableau via Novelli-Pak-Stoyanovskii

Let $T_{\lambda}$ be the set of standard young tableaux (SYT) of shape $\lambda_1\geq \lambda_2\cdots\geq \lambda_n$. Now consider pushing a row $\mu$ with $\mu\geq \lambda_1$ onto $Y$ to give shape ...

**7**

votes

**1**answer

350 views

### Quest for a human proof of a $q-$binomial identity

Let $$f(n,k) = \sum\limits_{j = - k}^k {{{( - 1)}^{k - j}}}
\binom{n-j}{k-j}\binom{n+j}{k+j}.$$
Then $f(n,k)=\binom{n}{k}$
because it satisfies $f(n,k)=f(n-1,k)+f(n-1,k-1)$ and the obvious ...

**6**

votes

**1**answer

232 views

### A variant to the Hadwiger-Nelson problem

Consider the following graph $G=(V,E)$ where $V=\mathbb{R}^2$ and $E = \{\{x,y\}: x,y \in \mathbb{R}^2 \text{ and } |x-y|\in \mathbb{Q}\}$.
What is $\chi(G)$?
(This is a variant of the ...

**3**

votes

**1**answer

98 views

### Minimal family of k-sets containing all t-sets

Let $n \ge k \ge t \in \mathbb{N}$, and consider a universe $U$ of size $n$. Let $\mathcal{F}$ be a family of $k$-subsets of $U$, such that every $t$-subset of $U$ is contained in at least one member ...

**4**

votes

**1**answer

292 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 ...

**8**

votes

**3**answers

295 views

### Distinguishing combinatorial maps by their linearizations

Every (not-necessarily invertible) map $f$ from $[n]:=\{1,2,,,,.n\}$ to itself determines a linear map $L_f$ from ${\bf R}^n$ to itself that sends the basis vector $e_k$ to $e_{f(k)}$ for $1 \leq k ...

**2**

votes

**0**answers

129 views

### Ticket lottery — distributing $n$ tickets among $N$ people fairly

Suppose that I have $n$ tickets for an event that I want to distribute fairly among $N > n$ people. In this simple case, a lottery suffices. But suppose certain groups of people want to attend ...

**10**

votes

**3**answers

395 views

### Is a distributive lattice planar iff it admits no B3 sublattice?

A finite lattice is planar if it admits a Hasse diagram which is a planar graph (we want the Hasse diagram to be represented in the plane so that the $y$-coordinate of each element respects the ...

**3**

votes

**0**answers

82 views

### Efficiently computing (plethysm-like?)substitutions of symmetric functions

This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...

**2**

votes

**1**answer

95 views

### Stirling numbers of the second kind with maximum part size

The stirling number of the second kind $S(n,k)$ counts the number of partitions of the set $[n]$ into $k$ non-empty parts. I found a definition for the numbers called the $r$-associated stirling ...