# Tagged Questions

**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

83 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 ...

**5**

votes

**2**answers

164 views

### Number of Dyck paths with prescribed number of edges

I am trying to find a formula for the trace of certain matrix. To do that I was forced to determine the number of Dyck paths with prescribed number of edges.
By a Dyck path I mean a lattice path from ...

**4**

votes

**1**answer

109 views

### Countable hypo-hamiltonian graph

If $G = (V,E)$ is a graph, then a $\omega$-path is an injective map $p:\omega\to V$ such that $\{p(k),p(k+1)\}\in E$ for all $k\in \omega$. In a similar fashion, we define a $\mathbb{Z}$-path.
Is ...

**4**

votes

**1**answer

210 views

### Counting couples of square-free polynomials over finite fields

I have a curve defined by the following equations over the finite field $\mathbb{F}_q$ with $q=p^r$ with $p \geq 3$:
$$C_{h_1,h_2}:\begin{cases} y_1^2=h_1(t) & \\
y_2^2=h_2(t) ...

**3**

votes

**2**answers

204 views

### Picking codewords that are close

I posted this question in http://math.stackexchange.com/questions/1142698/picking-codewords-that-are-close a week back.
Let $[n,k,d]$ be a linear code over $\Bbb F_q$ with minimum distance $d$ and ...

**2**

votes

**0**answers

74 views

### Exact growth rate of Longest Increasing Subsequence expectation

Let $S_n$ be the symmetric group, $\pi\in S_n$ a uniformly random permutation and $L_n:=L_n(\pi)$ denoting the length of the longest increasing subsequence (LIS). We know that ...

**3**

votes

**1**answer

269 views

### How many hamiltonian paths can be removed from a complete directed graph before it becomes disconnected?

The question started from a problem brought home by a friend's 5th grader: "How many ways you can arrange 5 people sitting around a round table so that the people sitting to the left of any person are ...

**0**

votes

**1**answer

79 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

**3**answers

88 views

### Block error-correcting codes over inhomogeneous alphabets

For $n := (n_1,\dots,n_N) \in \mathbb{N}_{>1}^N$, let $X_n := \prod_{j=1}^N [n_j]$, where as usual $[m] := \{1,\dots,m\}$.
Are there any known generic constructions for (Hamming) sphere ...

**5**

votes

**1**answer

161 views

### English translation of Witt's paper on the Mathieu groups?

Does anyone know of an English (or French) translation of Witt's paper Die 5-fach transitiven Gruppen von Mathieu ? (It's the one in which the Witt design is introduced. Well, I guess.)
Here's an ...

**2**

votes

**1**answer

107 views

### Network flows with shared capacities

Suppose we have a flow network, with capacity constraints on weighted sums of arc flows, such as:
$$2 f(1, 2) + 3 f(4, 5) + f(3, 7) \leq 10,$$
where $f(1, 2)$ denotes the flow through arc $(1, ...

**1**

vote

**0**answers

124 views

### Counting number of ways to place bags and marbles inside bags so all bags contain an odd number of marbles

There was a famous trick question posed once in math.stackexchange here. The question can be loosely translated to "Is it possible to place nine marbles into four bags so that each bag has an odd ...

**6**

votes

**1**answer

200 views

### A combinatorial identity generalizing identity (3.111) from Gould's book

I am trying to prove the following identity, which I am sure is correct:
$$
\sum_{m=0}^{K}(-1)^m{n \choose m}{(n-m)r-1 \choose n-1}=1,
$$
where $K:=\left[n\frac{r-1}{r}\right]$ for some integer $r$, ...

**3**

votes

**1**answer

264 views

### Number of semi-standard tableau

What is the number of semi-standard tableau (weakly increasing on rows and strictly increasing on columns) for the partition $2n=n+n$ with entries $\{1,2, \cdots ,n\}$ such that each $i$ appears ...

**0**

votes

**1**answer

152 views

### Number of squares in a grid under certain conditions

Consider an $(n+1)\times (n+1)$ grid of lattice points in the plane.
$A(n):$ # of squares with vertices on the grid.
It's relatively well-known that $A(n)=\frac{n(n+1)^2(n+2)}{12}$. Now, $A(n) = ...

**3**

votes

**3**answers

362 views

### Waring problem for binomial coefficients (generalization of Gauss' Eureka Theorem)

Is there a number k such that every natural number can be written as $\sum_{i=1}^k \binom{a_i}{3}$ for some natural numbers $a_i$'s?

**1**

vote

**0**answers

97 views

### Occupancy problem with limited capacity and two types of balls [closed]

I am considering the following problem that I suspect to be standard.
One has a set of $N$ balls composed of a fraction $\alpha$ of red balls and $(1-\alpha)$ of black balls (we assume $\alpha N$ is ...

**3**

votes

**0**answers

223 views

### Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...

**3**

votes

**0**answers

69 views

### Where does this identity involving sums of Hankel-like determinants over partitions come from?

For a partition $\lambda=( \lambda_1,\dots,\lambda_n)\vdash n$ with $\lambda_1\ge\dots\ge\lambda_n\ge0$ and any function $f:\mathbb Z\to\mathbb C$, define a Hankel-like $n\times n$ matrix ...

**12**

votes

**0**answers

310 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 ...

**12**

votes

**0**answers

263 views

### Grothendieck on polyhedra over finite fields

In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular ...

**4**

votes

**0**answers

57 views

### Is every planar point set be projections of vertices of a neighborly 4-polytope?

More exactly, written in coordinates, I'm curious to know if for every point set $(x_i,y_i)$ there are $(x_i,y_i,z_i,w_i)$ be vertices of a neighborly polytope.
This problem comes from a simple ...

**5**

votes

**1**answer

119 views

### Beraha numbers and zeros of the chromatic polynomial of planar graphs

Question: What is the largest Beraha number known to be an accumulation point of real zeros of the chromatic polynomial of planar graphs?
Background:
The Beraha numbers $B_n=2+2cos(2\pi/n), ...

**1**

vote

**0**answers

121 views

### counting how many boxes from a given Young tableau contribute to hook length made out of two YTs

Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...

**10**

votes

**1**answer

278 views

### Dynamics of RSK

There is a way of viewing the RSK correspondence as a map (in fact, bijection) $A \overset{RSK}\longrightarrow \widehat{A}$ from $n\times n$ matrices with entries $\mathbb{N}$ to (weak) reverse plane ...

**5**

votes

**0**answers

116 views

### LS paths construction

Let $W$ be the Weyl group of a simple Lie algebra $\mathfrak L$, and for a dominant weight $\lambda$ denote by $W_{\lambda}$ the stabilizer of $\lambda$ in $W$. Let $\leq$ be the Bruhat order on ...

**3**

votes

**2**answers

202 views

### Number of subsum of a given set of integers

I am asking myself for few days the following but I can't find any references, although I am pretty sure this was already studied, because it seems quite natural:
Given n integers $(e_i)$ chosen ...

**2**

votes

**1**answer

145 views

### Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric).
Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix.
Case $1$: $M+W\in\{0,1\}^{n\times n}$.
Could ...

**2**

votes

**1**answer

86 views

### Complex lines arrangements from a given wiring diagram

I'm working on some presentation of the fundamental group of the complement of a complex lines arrangement in $\mathbb{C}^{2}.$ In particular, in Arvola's article
"The fundamental group of the ...

**3**

votes

**1**answer

122 views

### Directed Hypercube Minimal Cuts

If $[n]:=\{1,2,\ldots, n\}$ for some $n\in\mathbb{N}$, then the hypercube digraph of dimension $n$, denoted $H_n$, is the graph whose set of vertices is the power-set $\wp([n])$ where two vertices ...

**0**

votes

**0**answers

67 views

### Decomposing matrices to lower ranks

Given real matrix $M\in\{0,1\}^{n\times n}$ of rank $r$. How many $M_i\in\{0,1\}^{n\times n}$ of rank $s$ does one need to write $M'=\sum_{i=1}^ta_iM_i$ for some $a_i\in\Bbb R$ where maximum absolute ...

**3**

votes

**1**answer

122 views

### How many expressions can be formed with $k$ commutative and associative functions?

This is a generalization of a question I posted to Math.SE here.
Suppose we have $k$ binary functions $f_1, f_2, \ldots, f_k$, all of which are commutative and associative, and $n$ indeterminates ...

**17**

votes

**1**answer

538 views

### A congruence involving binomial coefficients

The following open problem was shown to me by Maxim Kontsevich. I
state it in a different but equivalent form. Let $a(n)$ be the sequence
at http://oeis.org/A131868, that is,
$$ a(n) ...

**3**

votes

**1**answer

136 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 ...

**5**

votes

**0**answers

185 views

### Find subset of collection of sets whose intersection has minimum average value

Let $a_1,\ldots,a_n>0$, and let $S_1,\ldots,S_d\subset\{1,\ldots,n\}$ (all non-empty).
For any $I\subseteq\{1,\ldots,d\}$, define $S(I)=\bigcap_{i\in I} S_i$.
Given some $1\leq s < d$, consider ...

**3**

votes

**1**answer

100 views

### A multinomial-type sum over compositions of an integer

I find myself needing to compute (or asymptotically estimate) the following sum over the $2^{S-1}$ compositions of $S$. I am hoping an expert in combinatorics (I am a computer scientist) will ...

**11**

votes

**3**answers

363 views

### Sets of points containing permutations - a Ramsey-type question

The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with ...

**1**

vote

**0**answers

177 views

### Polynomial existence over finite field

Denote $\mathcal{F_n}$ as collection of multiaffine polynomials $f\in\Bbb F_2[x_1,\dots,x_n]$.
Denote total degree of $f\in\mathcal{F_n}$ as $deg(f)$ (note $deg(f)\leq n$).
Denote ...

**3**

votes

**1**answer

204 views

### Combinatorics problem involving counting the number of certain substrings

I'm not sure if this question is suited for MO, but it does seem quite challenging to me, and is required for a research problem in chemistry I'm working on. I did try getting help from elsewhere ...

**17**

votes

**2**answers

782 views

### A possibly surprising appearance of Lucas numbers

Let $S$ be the set of polynomials defined as follows: $0$ is in $S$, and if $p$ is in $S$, then $p + 1$ is in $S$ and $x \cdot p$ is in $S$, so that $S$ "grows" in generations: $g(0)=\{0\}$, ...

**2**

votes

**1**answer

91 views

### When the vertex covering number is smaller than the chromatic number

For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$.
As noted here, we have $\tau(G) \geq \chi(G) - 1$ for all finite graphs $G$. I'm interested in graphs $G$ ...

**8**

votes

**2**answers

824 views

### Surveys of the items of Erdős' “toolbox”

Could you point out some survey papers and monographs that highlight the kernel of tricks, techniques, and tools that Paul Erdős employed the most in his research work (in particular in graph theory, ...

**1**

vote

**0**answers

32 views

### Optimization of a multilinear function over a product of hypersimplices

Let $P = \Delta_1 \times \cdots \times \Delta_N$ be the Cartesian product of $N$ hypersimplices. Let $f : P \to \mathbb{R}$ be a multilinear function of $N$ variables, ie $x_i \mapsto f(x_1, \ldots, ...

**4**

votes

**3**answers

100 views

### Linear intersection number and vertex covering number

A linear hypergraph is a pair $\pi=(X, L)$ where $X\neq \emptyset$ is a set and $L\subseteq {\cal P}(X)$ has the following properties:
for $e\in L$ we have $|e|\geq 2$;
if $e_1\neq e_2 \in L$ then ...

**4**

votes

**0**answers

82 views

### Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries).
Can we partition this union into at most $n$ rectangles?
I think it's pretty ...

**0**

votes

**0**answers

51 views

### About adjacency matrices of $k-$shift lifts of graphs

I am finding the notation of cyclic lifts of graphs to be very confusing.
Lets say one is looking at a cyclic $k-$lift of a $\vert V \vert$ sized graph.
I would like to understand what is the ...

**3**

votes

**0**answers

53 views

### Signatures of latin squares: what about the extremal cases?

For a latin square (LS) of order $n$, we will define a cut (or maybe general transversal, I don't know whether there is an entrenched name for this) as a collection of $n$ cells such that no two share ...

**4**

votes

**1**answer

185 views

### Continuous-piecewise-linear versus piecewise-linear

Some authors use the term "continuous piecewise-linear" where other authors use the shorter term "piecewise-linear" (with continuity tacit).
I'd be interested in people's thoughts about this ...