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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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
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1answer
39 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 ...
2
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0answers
33 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 ...
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0answers
16 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
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0answers
32 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
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1answer
169 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 ...
4
votes
1answer
66 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 $x_i\in D$ ...
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0answers
75 views

Probabilistic statement on matrix ranks

Given $A\in\{0,1\}^{n\times n}$ with $rank(A)=r=2^{O(\sqrt{\log_2n})}$. Denote $\mathsf{1_n}\in\{0,1\}^n$ as vector with $1$s. Does $$\lim_{n\rightarrow\infty}\mathsf{P_{A\in\{0,1\}^{n\times ...
4
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0answers
120 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
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2answers
335 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 ...
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2answers
210 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
1answer
134 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 ...
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2answers
233 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 ...
5
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2answers
267 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
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0answers
65 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
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0answers
83 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
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1answer
75 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 ...
9
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2answers
317 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 ...
16
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429 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 ...
5
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1answer
227 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 ...
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0answers
59 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
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1answer
318 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
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1answer
110 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. ...
4
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1answer
120 views
+100

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
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0answers
54 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
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1answer
105 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 ...
-1
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1answer
79 views

Maximal independent sets in a graph $G$ versus maximal matchings in the line graph $L(G)$

I'm a bit confused because of the answers in Maximum matchings in infinite graphs . I was thinking that an independent set in a graph $G$ corresponds to a matching in the line graph $L(G)$, and vice ...
3
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0answers
53 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 ...
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35 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
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2answers
286 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
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0answers
69 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 ...
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64 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
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1answer
247 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 ...
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0answers
33 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 ...
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0answers
62 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 ...
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0answers
86 views

Martingales and Bipartite graphs

Would a vertex exposure martingale be useful for bounding the deviation in size of the largest matching from it's expected value in the standard random bipartite graph with vertex classes of size $n$ ...
2
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1answer
114 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?
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0answers
86 views

Determining strong base-orderability of a matroid

A matroid is said to be strongly base-orderable when for any two bases $B_1,B_2$ there exists a bijection $f:B_1 \mapsto B_2$ such that for any $X\subseteq B_1$ set $B_1 - X+ f(X)$ is also a base. ...
2
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1answer
176 views

Minimally intersecting subsets of fixed size

The question I have, is how to generate the following collection of subsets: Given a set $S$ of size $n$. I want to find a sequence of $k$ subsets of fixed size $m$, $0<m<n$, such that at each ...
5
votes
1answer
196 views

Is there a standard name for this poset

I've run into the following poset and I would expect it has a standard name. Let $n\geq k\geq 0$. Then $P_{n,k}$ consists of all $k$-element subsets of $\{1,\ldots,n\}$ ordered by $X\leq Y$ if ...
2
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2answers
130 views

Expected matching in a bipartite graph

Consider a random bipartite graph constructed on vertex classes of size $n$ with each edge present independently with probability $p$. How could I go about calculating the size of the expected ...
5
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1answer
271 views

Common sizes of intersections

I'm trying to come up with the largest family of sets that obeys the following properties: Consider $X = \{1,\dots,n\}$ and take $\mathcal{F} \subset 2^X$ such that for any three subsets $A,B,C \in ...
2
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0answers
103 views

Number of degree $k$ functions [closed]

Given a Boolean function $f:\{0,1\}^n\rightarrow\{0,1\}$, there is a real multivariate multilinear polynomial that is associated with in through interpolation. Example: ...
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2answers
173 views

A specific polynomial triplet question

Notation $P_k[n]=\{$multilinear polynomials in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$ of total degree exactly $k\}$. $k=1$ is just linear polynomials. QUESTION Is there a triplet $(p,f,g)\in ...
5
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0answers
51 views

Covering a set with images of a transversal

Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$. By a transversal we mean a set $\lbrace\omega_1,\ldots,\omega_k\rbrace$ with $\omega_j\in\Omega_j$ for ...
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0answers
61 views

Rook Polynomial of Quasi-Ferrers Board?

One can compute the rook polynomial of the following board: by transforming it to the following equivalent board, which is a Ferrers board, and then using the formula given here: How to compute ...
5
votes
6answers
548 views

Binomial coefficient identity

It seems to be nontrivial (to me) to show that the following identity holds: $$ \binom {m+n}{n} \sum_{k=0}^m \binom {m}{k} \frac {n(-1)^k}{n+k} = 1. $$ This quantity is related to the volume of the ...
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41 views

Embedding a collection of finite subsets efficiently

Are there any general non-trivial methods for solving the following problem? Suppose one has a collection of subsets $\mathcal{C} \subseteq \mathcal{P}\mathcal{P}\{1,\dots,n\}$. They may be viewed ...
2
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1answer
91 views

Minimality condition in a certain class of hypergraphs

A hypergraph is a pair $H=(V,E)$ such that $V$ is a (possibly infinite) set and $E\subseteq \mathcal{P}(V)$. $C\subseteq E$ is said to be a cover if $\bigcup C = V$ and $C$is minimal if $C'\subseteq ...
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0answers
135 views

covering high dimensional hypercube by balls

suppose we are given the $d$-dimensional hypercube $H^d$ defined as $$ H^d:=\left\{\sum_{i=1}^d\epsilon_ie_i:\ \epsilon_i\in \{0,1\}\mbox{ for }i=1,\dots , d\right\} $$ and $(e_i)_{i=1}^d$ the ...