**1**

vote

**1**answer

82 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 ...

**1**

vote

**1**answer

63 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

95 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

64 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

118 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 ...

**16**

votes

**1**answer

426 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

101 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?

**5**

votes

**1**answer

140 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

93 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

351 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

163 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 ...

**1**

vote

**1**answer

128 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

760 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

87 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 $G$ ...

**7**

votes

**2**answers

802 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

28 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

94 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

77 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

39 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

50 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

184 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 ...

**6**

votes

**1**answer

133 views

### Compiling self-referential forms

Fix $1\leq d\in\mathbb{N}$ and set $D:=\{0,1,\ldots,d-1\}$.
Consider the system of equations
\begin{equation}
x_i=c_i + \sum_{j\in D}\delta_{x_j,i}
\end{equation}
with $c_i\in D$ given and $x_i\in D$ ...

**1**

vote

**1**answer

182 views

### Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $\operatorname{rank}(A)=r=2^{O((\log_2n)^{\frac{1}{c+1}})}$.
Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s.
Does
...

**4**

votes

**0**answers

144 views

### Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...

**15**

votes

**2**answers

388 views

### Can all unit-distance graphs have their vertices at algebraic integers?

A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...

**8**

votes

**2**answers

233 views

### Certain signed sum over $S_n$

The following question appeared in my research:
Let $G_1,G_2,G_3$ all be subgroups of $S_n$, and consider the sum
$$
\sum_{g_i \in G_i, g_1g_2g_3 = id} \epsilon(g_1)
$$
that is, we only consider ...

**7**

votes

**1**answer

164 views

### A positivity problem involving the number of ways of expressing $n$ as a product of $k$ factors

Let $d_k(n)$ denote the number of ways of expressing $n$ as a product of $k$ factors, and let $$D_k(x)=\sum_{n\leq x}d_k(n)$$ be the summatory function. During a study of Mertens' function I was lead ...

**0**

votes

**3**answers

295 views

### On independent sets of graph

Given $G$ a regular graph on $n$ vertices, denote $\alpha(G)>1$ to be independence number.
Denote $\Gamma(G)$ to be collection of possible subset of independent vertices in $G$ of cardinality ...

**6**

votes

**3**answers

465 views

### Identity for Power Series and Binomial Coefficients

This question concerns a combinatorial identity obeyed by power series coefficients. Throughout we let $[x^{M}]\{\phi(x)\}$ denote the coefficient of $x^{M}$ in a power series $\phi(x)$.
Let $k$ be ...

**2**

votes

**0**answers

80 views

### Asymptotic expansion for the Bell numbers

The Bell numbers $B(n)$ (that is, the numbers that count the set partitions of a set, and have exponential generating function $\exp(e^x -1)$ ) admit the asymptotic expansion
$$\frac{\log B(n)}{n} = ...

**2**

votes

**0**answers

93 views

### Isomorphic subcategories of directed graphs and presets

For the purposes of this post, a digraph (directed graph) has neither loops nor multiple parallel edges, and a preset is an ordered pair consisting of a set $S$ and a preorder (viz., a reflexive and ...

**2**

votes

**1**answer

88 views

### Counting linearly ordered subsets of maximal length in partially ordered $d$-tuples of nonnegative integers

Given $d\in \mathbb{N}$, let $X_d:= \{(\ell_1, \ldots , \ell_d): 0 \le \ell_1 \le \ldots \le \ell_d \le d\}\subset \mathbb{Z}^d$, and endow $X_d$ with the (usual) partial order, namely, $x\le y$ if ...

**10**

votes

**2**answers

348 views

### Can a graph be reconstructed from its cycle lengths?

All graphs discussed are finite and simple. The cycle sequence of a graph $G$, denoted $C(G)$, is the nondecreasing sequence of the lengths of all of the cycles in $G$, where cycles are distinguished ...

**18**

votes

**1**answer

633 views

### What are the products $\prod_{A\subset{\mathbb F}_p\colon |A|=n} \sum_{a\in A} \zeta^a$ equal to?

This is a somewhat more explicit version of a question I have recently asked.
Let $p$ be an odd prime, and write $\zeta:=\exp(2\pi i/p)$ (any other primitive $p$th root of unity will do as well). For ...

**6**

votes

**1**answer

246 views

### Infinite graphs isomorphic to their line graph

The only finite connected graphs $G$ that are isomorphic to their line graph $L(G)$ are the cycle graphs $C_n$ (see this link for example).
There are connected countable graphs that are isomorphic to ...

**1**

vote

**0**answers

65 views

### research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$

Is there any research on the structure/properties of permutation matrix/table with $(i,j)th$ entry as $\pi_j\circ \pi_i^{-1}$, where $\{\pi_1,\pi_2,...,\pi_{k!}\}=S_k$?
I know if we apply the ...

**2**

votes

**1**answer

349 views

### Is it possible to create an infinite sequence in which no subsequence is repeated 3 times in a row?

In Chess, there is the Threefold Repetition rule where if a sequence of moves is repeated 3 times in a row, either player can claim a draw.
Say two players wanted to play a legal, infinite game of ...

**3**

votes

**1**answer

126 views

### Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams.
I also found some relation with matroid theory. ...

**6**

votes

**1**answer

221 views

### Induced subgraphs of small strongly regular graphs

Consider a strongly regular graph $G$ with parameters $(76,30,8,14).$ Hoffman's bound tells us that $\overline{G}$ has an independent set of size at most $4$ and its not hard to see there are indeed ...

**4**

votes

**0**answers

65 views

### Lower bound on the number of k-plexes in a Latin square

Let $A$ be an order-$n$ Latin square. A $k$-plex of $A$ is a set of entries , $k$ from each row and column and $k$ from each symbol.
My question is: Is there a Latin square with a large number of ...

**3**

votes

**1**answer

108 views

### Infimum of partitions

Let $X\neq\emptyset$ be a set. A partition is a subset $P\subseteq {\cal P}(X)\setminus \{\emptyset\}$ such that $\bigcup P = X$ and any distinct members of $P$ are disjoint. We denote by ...

**3**

votes

**0**answers

76 views

### Nonattacking configurations of k bishops on an m by n rectangular board

The number of ways to place k bishops in a nonattacking configuration on an n by n square board is a well known result and can for example be found in ...

**0**

votes

**0**answers

40 views

### Number of faces of polytope projecting to lower dimensional polyhedron

Denote $K=\mathrm{conv}(v_1, \ldots, v_n)\subsetneq\Bbb R^m$ to be convex set spanned by vectors $v_i\in\Bbb R^m$ with $m\leq n$ then what technique could be useful to upper bound minimum number of ...

**3**

votes

**2**answers

297 views

### How to flip one triangulation on a surface into another

Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$.
Is there an algorithm which takes two ...

**2**

votes

**0**answers

94 views

### Volume of bounded regions in hyperplane arrangements

I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors ...

**1**

vote

**0**answers

80 views

### Multiple Bipartite graphs and matchings

I've been told recently that it's better i just for help regarding my 'specific' problem rather than lots of little questions around the same topic which appear somewhat unclear. I would first like to ...

**10**

votes

**1**answer

267 views

### Soft question: mathematics about truchet tiles

It seems that this is the first question on Truchet tiles on MO.
Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.
I ...

**1**

vote

**0**answers

34 views

### Rook Polynomials of Skew-Ferrers Boards

What are some known method for calculating the rook polynomials of skew Ferrers boards? Currently all I have been able to find is the following paper Bruhat intervals as rooks on skew Ferrers boards ...

**0**

votes

**0**answers

63 views

### When is edge colored circulant isomorphism polynomial?

Don't understand enough group theory, but two papers
appear to give partial results about an open problem.
Edge colored graph isomorphism is isomorphism which
preserves the edge coloring (the ...

**2**

votes

**1**answer

127 views

### NP-hardness of finding maximum of minimum element in diagonal of a matrix

For $A = \{a_{ij}\} \in R^{n\times n}$, is finding
$$
\max_{\sigma \in S_n}\min_{1 \le i \le n} a_{i,\ \sigma_i}
$$
NP-hard?