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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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 ...
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115 views

Detecting Bijection between two Permutation Set [on hold]

Let, $L$ is a set of $n$ labels / colors (repetition of label/color is possible). Assume, there is a function $f$, that maps labels of $L$ to itself. this mapping is bijective. Let, $\beta$ is a set ...
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1answer
73 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 ...
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1answer
84 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 $\mathcal{F}$ ...
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149 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, ...
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3answers
188 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, ...
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1answer
232 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 ...
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56 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 ...
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1answer
104 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. ...
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51 views

Central limit theorem for perfect matching counts [closed]

This is a modification to one of my questions: Set $N_G$ the number of copies of graph $G$ in the Erdős–Rényi random graph model $G(n,p)$. We have the law of large number for the number of copies of ...
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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 ...
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66 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 ...
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1answer
81 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 ...
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31 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 ...
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1answer
195 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 ...
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1answer
55 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 ...
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1answer
396 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 ...
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144 views

A conjecture about orbits in a recursive function motivated by Kolakoski's sequence

We first introduce several functions motivated by Kolakoski's sequence. The conjecture itself can be stated independently of Kolakoski's sequence. You can skip straight to the formulation of the ...
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1answer
313 views

Combinatorics problem about sum of natural numbers

Following combinatorics problem is claimed to be an open problem in "The Princeton Companion to Mathematics" (pp. 6) Let $a_1,a_2,a_3,...$ be a sequence of positive integers, and suppose that each ...
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2answers
109 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 ...
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1answer
235 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 ...
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1answer
139 views

Can we deduce that a finite topology $T$ satisfies Frankl's union-closed set conjecture?

Let $X$ be a finite set and $T$ be a topology on $X$. Then $T$ is both union-closed and intersection-closed. Can we deduce that $T$ satisfies Frankl's union-closed set conjecture? (We know that a ...
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36 views

Separation on discrete set

Consider the set $L = \prod_{i=1}^n\{1,0\}$, i.e. L consists of the element of n-tuples whose entries are 0 or 1. Also we can regard $L$ as a subset of $R^n$. Define linear functions $f(x)= a_1x_1+ ...
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1answer
91 views

Probability of existence of a base in the span of sparse vectors in GF(2)

For $i=1,2,\dots,l$, let $\mathbf{v}_i =(v_{i1},v_{i2},\dots,v_{in}) \in \mathbb{F}_2^n$ be a sparse vector in GF(2) such that all $v_{ij}$'s are independent for all $1 \le i \le l, 1 \le j \le n$ and ...
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1answer
155 views

The class of $(-1,0,1)$-matrix with all row sums and column sums equalling to $0$

Let $n$ be an even positive integer and $W_n$ be the class of all $n\times n$ matrices with entries from the set $\{-1,0,1\}$ satisfying all row sums and column sums are equal to $0$. For any $M\in ...
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1answer
87 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 ...
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2answers
133 views

Characterizing graphs whose Incidence Matrix has the all ones vector in its row span

Suppose we have a simple connected graph $G=(V,E)$. Then let $A$ be its $|E|\times |V|$ incidence matrix. Here I am considering the unoriented incidence matrix. I want to known when the row span of ...
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0answers
68 views

Given a vector of positive integers, count the number of combinations which have a sum that produces a different value

I have a list (vector) of positive integer numbers, including repetitions. For example, $L = [1, 1, 4, 1, 3]$; I want to calculate the number of different sums obtained by using the elements of $L$, ...
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55 views

bounded degree graph colouring.

I was wondering if anyone could provide references on the following: Is determining the chromatic number of a bounded degree graph APX-complete? 2.I've seen the result that states it is NP-hard ...
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45 views

Hypergraph edge colouring

I'm interested in knowing if finding the edge-chromatic number of a $k$-uniform $k$-partite hypergraph is NP-hard for $k\geq 3$ Could anyone provide a reference for the result? By edge-chromatic ...
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1answer
80 views

Choice number of embedded graphs

For given $g$, consider the family of graphs which may be embedded to the compact orientable surface of genus $g$. For this family, consider maximal clique $\alpha(g)$, maximal chromatic number ...
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1answer
224 views

upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here. You can skip examples below and read from "General setting" at ...
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1answer
203 views

Can we cover a set by a particular family of sets?

Let $A_1,A_2,\ldots,A_m,B_1,B_2,\ldots, B_m$ be (not necessarily distinct) subsets of $[n]=\{1,2,\ldots,n\}$. Suppose that each $i\in [n]$ appears in at least $k$ of these $2m$ sets. I want to ...
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1answer
85 views

Weighted counting of circular codes

Given a circular code $X$ (for example: $X=\{ w,b \}$) with generating function $u(z)=\sum\limits_{k=0}^{\infty}{u_k z^k}$ (in this example : $u(z)=2z$), the generating function ...
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1answer
173 views

A spectral graph theory problem

Let $S$ be a zero-free subset of the group ${\bf Z}_2^n$ and $\Gamma={\rm Cay}({\bf Z}_2^n,S)$ be a bipartite Cayley graph. For some choices of $S$, the graph $\Gamma$ has $4$ distinct eigenvalues, ...
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57 views

Terminology and technique for repeated pairwise removal of elements of posets: “Collapsibility” of a “face poset”

Let $P$ be a poset, or partially ordered set. Let $\le$ denote the reflexive order on $P$, and $<$ the corresponding irreflexive order. Let the phrase "a maximal pair" in $P$ refer to an ordered ...
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1answer
65 views

Separate a special poset by function

Assume $A = \prod_{i=1}^n\{0,1\}$, i.e. element $(a_1,\cdots,a_n)=a\in A$ is n-tuples like $(1,0,1,\cdots)$. There is an obvious partial order on the $A$: say $a < b$ for $a,b\in A$ if and only ...
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63 views

Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
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1answer
122 views

Union-closed family generated by n 2-sets

I asked this question on Stackexchange, but I got no answer, so I ask it here. Let us define a $2$-set as a set with exactly $2$ elements. For a natural number $n$, let $l(n)$ denote the least ...
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62 views

Lower bound the ratio of the combinatorial quantities

Suppose $p < q$, $s = p^{d}$ for some fixed $d \in (0,1)$, let $p$ goes to infinity, define the following quantity, \begin{aligned} \quad f(j) = \sum_{i = 0}^{\min(j,s)}{s \choose i}{p-s \choose ...
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1answer
157 views

sum over all integer partitions, of the product of the factorials of the terms

I'm looking for something making tractable the sum, over all partitions into k terms of an integer n, of the product of the factorials of all the terms. Thanks,
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connectedness of semi algebraic sets

We know the inequalities $x_ix_j >\theta_{ij}$ or $x_ix_j<\theta_{ij}$ for some $\theta_{ij}$>0, some $i,j\in\{1,\cdots,n\}$, $i\neq j$ defines the easiest semi algebraic set in $R^n_{\geq 0}$, ...
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1answer
106 views

Average minimum number of random k-sparse vectors in GF(2) to span the whole space?

What is the average minimum required number of independent $k$-sparse (having at most $k$ non-zero elements) random vectors belonging to $\mathbb{F}_2^n$ to span the whole space of $\mathbb{F}_2^n$? ...
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1answer
99 views

What is the densest bipartite graph with unique Hamiltonian cycle?

In a prior post regarding perfect matching, it was stated that the densest graph with a unique perfect matching cannot have more than $n^2$ edges, if graph has $2n$ vertices. Analogously, what is the ...
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1answer
132 views

Upper Bound for the Difference of Even Probability and Odd Probability in Hypergeometric Distribution

Let $X$ be a random variable following the hypergeometric distribution with parameters $N,K,n$, where \begin{equation} Pr(X=k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}. \end{equation} To ...
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First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...
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1answer
131 views

Number of eigenvalues of a Cayley graph

Let $G=Z_2^n$ and $S\subset G$. Is there any relation for number of distinct eigenvalues of $\Gamma=Cayley(G,S)$ graph depending on $n$ and $|S|$, or at least diameter of $\Gamma$? If you have any ...
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1answer
130 views

“Small” simplicial complex with torsion trees

I am giving an expository talk soon about Duval-Klivans-Martin's paper Simplicial Matrix Tree Theorems, and I've been struggling to find a good example to do at the board. An important aspect of the ...
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1answer
113 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 ...
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1answer
283 views

Choosing two-colorable subgraph in a triangulation

Consider a planar graph $G$ which is a triangulation. Is it possible to find a two-colorable subgraph $H$ of $G$ which has a common edge with every face of $G$? It is known that it is not always ...