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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5
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2answers
79 views

Graphs with prescribed numbers of k-cliques

Let $(a_1,a_2,\dots, a_n)$ be a sequence of non-negative integers. Q. When does there exists a simple graph $G$ such that its number of $k$-cliques is $a_k$ (that is $G$ has $a_1$ vertices, $a_2$ ...
6
votes
1answer
542 views

Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...
5
votes
2answers
599 views

Where does the definition of “Tower of Algebras” come from?

A tower of algebras is a sequence of algebras $$A_0 \hookrightarrow A_1 \hookrightarrow \cdots \hookrightarrow A_n \hookrightarrow \cdots$$ with embeddings $A_n \otimes A_m \hookrightarrow A_{n+m}$ ...
3
votes
1answer
126 views

Alternative parallel paths

There are $n$ non-intersecting strings (with ends $x_1,\dots, x_n$ and $y_1,\dots, y_n$). An additional string intersects the first $n$ strings somehow. All the intersections are simple (vertices of ...
2
votes
0answers
41 views

How many distinct sets of n collinear points are there in an evenly-spaced two-dimensional grid of m x m points?

I'm seeking the definition of some function $f(n,m)$ which evaluates to the number of distinct sets of $n$ collinear points which are selected from an evenly-spaced two-dimensional grid of $m \times ...
2
votes
1answer
100 views

How many lines of exactly n points can be placed in a discrete, square grid of size m x m?

Per the title, I'm seeking the definition of a function $f(n, m)$ which evaluates to the number of lines made from exactly $n$ points which can be placed on a two-dimensional discrete, square grid of ...
7
votes
3answers
612 views

How can we count lines in an n-x-n rectangular array?

Is there a formula for the number of lines that contain exactly two points through an n x n rectangular array of points?
5
votes
0answers
67 views

Multiplication in universal enveloping algebra in terms of PBW isomorphism

Let $\mathfrak g$ be a Lie algebra. Consider the multiplication map $m:\mathfrak g\otimes U(\mathfrak g)\to U(\mathfrak g)$ and $i:S(\mathfrak g)\to U(\mathfrak g)$ -- Poincare-Birkhoff-Witt ...
2
votes
0answers
185 views

$G$-equivariant coherent sheaves on Bott-Samelson Resolutions

Let $G$ be a Lie group, $B$ be a Borel subgroup. $G/B$ is the corresponding flag variety. Let $w$ be an element of the Weyl group $W$ with a reduced expression $w = s_1 \cdots s_n$. Let $X_w$ be ...
0
votes
0answers
67 views

Finding stable ideals of $\mathbb{F}_3[[X,S]]$ by group action

Let $k > 1$ be a positive integer and define the action $\sigma_k$ on $\mathbb{F}_3[[X,S]]$ by: $\sigma_k: X \mapsto X + S + X^k$ $\sigma_k: S \mapsto S + S^3$. Conjecture: There exists a ...
2
votes
0answers
81 views

Nonvanishing of the dual Euler totient on boolean intervals of finite groups

The rank $n$ boolean lattice $B_n$, is the subset lattice of $\{1,2, \dotsm n \}$. Let $[H,G]$ be a boolean interval of finite groups. Its Euler totient is defined by $$\varphi(H,G):=\sum_{K \in ...
0
votes
0answers
21 views

LP or IP necessary? Network Flow Problem with no cycle-condition (unimodularity?) [on hold]

I need your help with a optimization problem. Recap: Normal mincost flow networks optimization problems have a constraint matrix which is total unimodular. This is a nice feature since a linear ...
3
votes
0answers
58 views

How many unimodular lattices does it take to fill a cube with high probability?

Consider $C_a$ in $\Bbb Z^n$ a cube of height $a$ at origin in positive coordinates with one corner at origin. Consider the set $M_c$ of all unimodular matrices in $\Bbb Z^{n\times n}$ with each ...
0
votes
0answers
61 views

Derandomizing AP existence in $A\subseteq \{1,\ldots,N\}$ for $\delta(A) \geq 1/k$

In the answer to the mathoverflow question here, it was established that if we let $p$ be the probability of including point $v$ in $A\subseteq \{1,\ldots,N\}$ and this is done independently for all ...
8
votes
1answer
575 views

Bipartite Graphs arising from two k-partitions of a given Graph

Let $G$ be an $n$-chromatic connected graph. Let $(V_1, V_2, \cdots, V_n)$ and $(U_1, U_2, \cdots, U_n)$ be two partitions of $V(G)$ corresponding to proper n-colorings of $G$. Consider the bipartite ...
1
vote
1answer
85 views

Convexity of truncated expectation

Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$. Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
8
votes
1answer
667 views

A Zero-Multiplicity Problem Related to Foulkes' Conjecture

I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or ...
0
votes
0answers
155 views

Warren's Theorem

At the end of page 12 in this document Noga Alon mentions Warren's Theorem on sign patterns: tau.ac.il/~nogaa/PDFS/tools1.pdf Does anyone know of an intuitive explanation of the proof of it ? Also, ...
2
votes
1answer
79 views

Catalogs/numbers/constructions of non-isomorphic conference matrices

I am interested in complete catalogs of non-isomorphic conference matrices, similar to those of Hadamard matrices. Do such catalogs exist? If yes, then where could they be found, and what is an ...
15
votes
11answers
2k views

Chromatic number of graphs of tangent closed balls

The Koebe–Andreev–Thurston theorem gives a characterization of planar graphs in terms of disjoint circles being tangent. For every planar graph $G$ there is a disk packing whose graph is $G$. What ...
23
votes
2answers
2k views

Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order ...
1
vote
1answer
73 views

Counting bounded genus non-isomorphic graphs

What is the number of non-isomorphic $2n$ vertex balanced bipartite graphs of degree at most $d$ and genus $g$? I am most interested in $d\leq3$ and $g=0$.
3
votes
1answer
92 views

Neighbourhood of a word and Levenshtein distance

The Levenshtein distance or Edit distance $$ lev(U,V) $$ between two strings $U$ and $V$ over a finite alphabet $\Sigma$ of size $ \left| \Sigma \right| = \sigma ,$ is the minimal number of ...
4
votes
3answers
177 views

Relation between diametral path and regularity of a graph

Let $G(V,E)$ be a graph. A path whose length is equal to the diameter of a graph is called a diametral path. In a cycle graph every vertex has $2$ diametral paths. Now I need to prove that this: ...
1
vote
0answers
102 views

A probability question related to combinatoric problem

I am trying to solve a combinatoric problem. The problem is the following: There are A,B,C three types of people. There are totally N people arriving sequentially and make a choice between two boxes X ...
11
votes
1answer
954 views

Menger's theorem via matroids

Let $G=(V,E)$ be an oriented graph, $Y\subset V$ be some fixed set of its vertices. Call $A\subset V$ independent if there exist $|A|$ vertex-disjoint paths starting in $A$ and ending in $Y$. It is ...
1
vote
1answer
281 views

Rewriting a series $\sum_{n=0}^\infty \frac{1}{n!}(\Delta^\varepsilon)^n a_n$ in the form $\sum_{n=0}^\infty c_n \varepsilon^n$

I would like to rewrite the series $$\sum_{n=0}^\infty \frac{1}{n!}(\Delta^\varepsilon)^n a_n,$$ where $\Delta^\varepsilon=\sum_{k=1}^\infty \varepsilon^k b_k$, as a series in $\varepsilon$ ...
1
vote
1answer
85 views

Intersection of members in a separating union-closed family of sets

Tony Huynh gave a nice answer to a question I asked here : Number of members of a separating union-closed family whose universe has given cardinality The answer shows in fact that if $\mathcal{F}$ ...
12
votes
3answers
7k views

Solving a Rubik's cube via a series of randomly selected (quarter-turn) Singmaster moves

In July of 2010, Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge demonstrated (computationally) that a $3\times3\times3$ Rubik's cube, starting in an arbitrary configuration, can ...
6
votes
1answer
206 views

Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?

The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that $$ \|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 ...
1
vote
3answers
198 views

Existence of special graph

Let $G$ be a $n$-vertices graph and $\lambda_1$ is the largest eigenvalue of this graph. If $\lambda_1$ is an integer value, we can easily find the $\lambda_1$- regular graph with $n$ vertices. Now, ...
0
votes
1answer
59 views

Choosing directed subgraph in a triangulation

Consider triangulation $T.$ Is it always possible to choose such a subgraph $H$ of $T$ that has a common edge with every face of $T$ and can be directed in such way that indegrees of all vertices of ...
3
votes
1answer
63 views

Generating Uniquely k-Optimal Point Sets

This question is motivated by the observation that finding an optimal tour through a set of points in the Euclidean plane is especially simple, if the points are in convex configuration and, that the ...
1
vote
1answer
142 views

Finding many disjoint sub-trees with many leaves

Let $T$ be a rooted binary tree with $L$ leaves, and let $\ell$ be a natural number smaller than $L$. The question is what is the maximal number of disjoint rooted sub-trees with at least $\ell$ ...
3
votes
1answer
82 views

Does the Chen-Chvátal Conjecture on metric spaces hold for maximal lines?

A conjecture by Chen and Chvátal asks for the minimum number of induced "lines" in a metric space, in the same spirit as the De Bruijn–Erdős theorem. Though the statement of this problem on Douglas ...
2
votes
1answer
105 views

Number of members of a separating union-closed family whose universe has given cardinality

If I'm not wrong, it is easy to prove the following statement : If $n$ is a natural number $\leq 4$, if $\mathcal{F}$ is a union-closed family of nonempty sets, if the universe of $\mathcal{F}$ (i.e. ...
10
votes
3answers
1k views

When does $P(a−b)=0$ for $a≠b$ ensure $P(0)=0$? (Continued.)

As a natural (and expectable) extension of my earlier question: How large must be a set $A\subset F_2^n$ to ensure that if $P$ is a cubic polynomial in $n$ variables over the field $F_2$, ...
4
votes
1answer
149 views

What is the minimum number of independent sets for a graph with fixed numbers of vertices and edges?

Fix integers $V$ and $E_{\text{max}}$, and consider graphs $G$ with $V$ vertices and at most $E_{\text{max}}$ edges. What is the best lower bound that one can give on the number of distinct ...
4
votes
1answer
240 views

Vector with many non-zero coordinates

Given finite field $\mathbb{F}_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least ...
2
votes
0answers
58 views

Number of $(2n-1)$-edge-colorings of the complete graph $K_{2n}$

I just started reading about graph theory and have a question (which might be trivial). How many $(2n-1)$ edge colorings of $K_{2n}$ are there? A vaguer question: can I write $K_{4n}= K_4 + K_4 ...
4
votes
3answers
2k views

Proofs of Lower Bounds for Ramsey Numbers?

As a sort of dual question to this question, I am wondering what proofs people know of lower bounds on Ramsey numbers $R(k, k)$. I know of two proofs: there is Erdos's beautiful probabilistic ...
4
votes
2answers
246 views

Are all numbers from $1$ to $n!$ the number of perfect matchings of some bipartite graph?

Let $f(G)$ give the number of perfect matchings of a graph $G$. Consider set $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$. Consider collection of all $2n$ vertex balanced bipartite graph to be ...
1
vote
1answer
37 views

How to generate a Latin square $M'$ in the same main class as $M \in \mathrm{LS}(9)$ which agrees with $L$ in the most cells?

I'm brainstorming an idea for storing a compressed list of main class representatives of Latin squares of order $9$. One way to compress the list would be to store one Latin square $L_1$, and for $i ...
18
votes
1answer
1k views

Simplest form for sum of Binomial Expressions

How difficult is the problem of reducing the number of terms in a sum of binomial expressions? Formally: Given $a_1, a_2, a_3, … a_n$, and $b_1, b_2, b_3, ... , b_n$, where $a_i, b_i \in \mathbb{Z}$, ...
2
votes
2answers
115 views

Probability of no $k$ 1's in arithmetic progression in binary sequence of length $n$

It is well known [it's on Wolfram Mathworld, for example] that the probability of no runs of $k$ consecutive $1$'s will occur in a $\{0,1\}$-valued sequence of length $n$ is exactly equal to ...
0
votes
0answers
67 views

Missing count in number of perfect matchings

Let $f(G)$ give the number of perfect matchings of a graph $G$. Consider set $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$. Consider collection of all $2n$ vertex balanced bipartite graph to be ...
3
votes
0answers
32 views

Does the rank (=height) of a well partial order bound its type (=length, =stature)?

Terminology and context (This should all be standard, but is recalled because terminology sometimes varies, and also to put the question into perspective.) A partially ordered set is called ...
3
votes
1answer
102 views

Choosing $K$ “centers” from the space of permutations

Let $\Pi$ denote the space of all permutations of $\{1,\dots,n\}$, and let $d(\cdot,\cdot)$ be a metric on $\Pi$. Suppose I am given a large integer $K$ and I have to select $K$ permutations ...
12
votes
1answer
410 views

A combinatorial identity involving harmonic numbers

The harmonic numbers are given by $$H_n=\sum_{k=1}^n\frac{1}{k}.$$ Numerical calculation suggests $$ \sum_{k=1}^{n}(-1)^k{n\choose k}{n+k\choose k}\sum_{i=1}^{k}\frac{1}{n+i}=(-1)^nH_n. $$ I can not ...
5
votes
1answer
238 views

Combinatorics: set partitions of a poset

Let $\pi_n$ be the poset of all set partitions of $\{1,...,n\}$ ordered by refinement, $\sigma = \{B_1,...,B_k\}$ be a set partition with blocks $B_i$, and $\max(B_i)$ be the maximum value in the ...