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Questions tagged [co.combinatorics]

Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

2
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2answers
831 views

Invariants of Matrix Reordering

are there any invariants of matrices, that are not affected by row- and/or column permutations? To me it seems that the sequence of singular values could be such an invariant; am I right, resp. are ...
2
votes
1answer
144 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 $n \leq 4$ is a natural number, if $\mathcal{F}$ is a union-closed family of non-empty sets, if the universe of $\mathcal{F}$ (i.e. ...
2
votes
1answer
98 views

Neighborhood maps for graphs $G$ with $\delta(G) \geq 2$

Let $G= (V,E)$ be a simple, undirected graph. For $v\in V$ we set $N(v) = \{w\in V: \{v,w\} \in E\}$. Note that we always have $v\notin N(v)$. A function $f:V\to V$ is said to be a neighborhood map if ...
2
votes
4answers
445 views

Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion.

Are there any results known about the discriminants of indefinite integral binary quadratic forms admitting automorphisms of order 3 or 6? It seems reasonable to expect that any permissible ...
2
votes
0answers
117 views

Link of a power series by the Bernoullis for a Riccati equation to zonotopes?

On pg. 85 of The Rise and Development of Theory of Series up to the Early 1820s by Ferraro is a series soln. of $$ d^2z/z = -x^2dx^2 $$ related to the reputed first appearance of a Riccati-type eqn.,...
2
votes
1answer
399 views

Proof that the $\omega$-language consisting of all words containing every finite word as a factor is not rational/regular

Let $\eta$ be an $\omega$-word over $X = \{0,1\}$ and let $F_k(\eta)$ denote the factors of $\eta$ of length $k$. Define the following $\omega$-languages $$ L_k := \{ \xi : F_k(\xi) = X^k \} = \{ \xi ...
2
votes
1answer
164 views

Asking for an example of a graph $G$ satisfying the following property

This question comes from another question I submitted earlier. Let $G$ be a finite graph. For any independent set $S$ in $G$ with $|S|\geqslant2$ and $v\in S$, define $$d_S(v)=\mid\{u\in V(G)\...
2
votes
2answers
315 views

Irreducible Polynomials from a Reccurence

This question is inspired by a recent one : Let $c$ be a variable and define a sequence by $a_0=0$ $a_1=1$ and $a_{n+1}=a_{n}c-a_{n-1}$ . So $$\begin{align*} a_2 &= c \\ a_3 &={c}^{2}-1= \...
2
votes
0answers
140 views

Sylvester-Gallai-type theorem for quadratic polynomials

Let $F_1, F_2$ and $F_3$ be finites sets of irreducible polynomials in $\mathbb{C}[x_0, \ldots, x_n]$ of degree at most $2$ such that $F_1 \cap F_2 \cap F_3 = \varnothing$ and for every $Q_1, Q_2$ ...
2
votes
1answer
115 views

Asymptotic behaviour of fixed points in permutations

For any $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
1
vote
0answers
66 views

Number of misplaced elements for a partition of a set of coloured items

We are given a set $V$ of $n$ items, where each item is tinted with exactly one color in $C=\{c_1, c_2, \ldots, c_k\}$, in such a way that for each $i\in [k]$ there exists at least one item tinted ...
1
vote
2answers
2k views

On the number of antichains of a poset

I am not an expert in combinatorics, but I need to count (or to approximate) the number of antichains of a poset. The idea relies on approximating this number by embedding my poset into another one, ...
1
vote
1answer
181 views

Infinite graph with degrees given

Let $\kappa$ be an infinite cardinal and suppose $$n, d: \kappa \to \big((\kappa+1)\setminus \{0\}\big) = \{1, \ldots, \kappa\}$$ are arbitrary functions. Is there $E \subseteq \big\{\{x,y\}: x\neq y ...
1
vote
1answer
449 views

Some divisibility constraints in Frobenius coin problem

Let's say that the linear form $ax+by$ represents $n$ if $ax+by=n$ for some positive integer $x$ and $y$. Call a pair $(a,b)\in\Bbb N\times\Bbb N$ with $\mathsf{gcd}(a,b)=1$ excellent if linear form ...
1
vote
2answers
144 views

A recursive Double sequence related to uniform Cardinal B-spline

Given a sequence $A_n(k)$ defined as follows: $A_0(0)=1$, $A_0(k)=0$ for all nonzero integers $k$ and $$A_{n}(k)=(n+1-k)^2A_{n-1}(k-1)+2(n(n+1)-k^2)A_{n-1}(k)+(n+1+k)^2A_{n-1}(k+1)$$ for all positive ...
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0answers
95 views

cospectral graphs have isomorphic adjacency matrices?

Let $G$ and $H$ be two cospectral graphs, then can we say that their adjacency matrices are isomorphic?
1
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1answer
283 views

Relationship of clique, independence, and chromatic numbers

For any graph $G=(V,E)$ let $\bar{G}$ be the complement graph. Is $$\text{inf}\big\{\frac{\omega(G)+\omega(\bar{G})}{\chi(G)} : G \text{ is a finite graph}\big\}$$ known? If not, what lower bounds are ...
1
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0answers
128 views

Indecomposability of image transformations (pure algebra). Open questions

W-transformations -- definitions We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
1
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1answer
651 views

Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by L....
1
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1answer
142 views

linear recurrence inequality of positive terms

This is a follow up on my previous linear recurrence inequality question. I have some matrices which satisfy a linear recurrence formula of the form $$ A_{n+1} = \alpha A_{n} + \beta A_{n-1},\qquad n\...
1
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2answers
46 views

Achieving every possible ranking by rearranging weights

This problem actually arose as a question in the real world (see the paragraph "Origin of the problem" below). Let $\mathbb{N}$ denote the set of the positive integers and let $n\in\mathbb{N}$. If $...
1
vote
1answer
315 views

Is the number of vertices bounded for fixed max degree and fixed diameter?

Are there positive integers $\Delta, d$ such that the following statement is true? For every $n\in \mathbb{N}$ there is a graph $G = (V,E)$ such that $|V| = n$, $\Delta(G) \leq \Delta$ (...
1
vote
4answers
383 views

All possible linear combinations of positive half-integers with coefficients +/- 1

This will be a simple problem on paper, but the brute force method is not really suitable for a computer, so I'm after a tricky algorithm that works in practice too: if $n$ positive half-integers $p_i$...
1
vote
2answers
164 views

Linear Extension of the $n\times n$ lattice

I am looking at a particular integer sequence, the number of $n\times n$ Young Tableaus (see OEIS). In the comments at OEIS, Mitch Harris stated that the same sequence also defines the Number of ...
1
vote
1answer
212 views

Odd permutations $\tau\in S_n$ with $\sum_{k=1}^nk\tau(k)$ an odd square

For any positive integer $n$, as usual we let $S_n$ be the symmetric group of all the permutations of $\{1,\ldots,n\}$. QUESTION: Is it true that for each integer $n>3$ there is an odd permutation ...
1
vote
1answer
199 views

Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory

Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let $(n_0,\...
1
vote
1answer
426 views

Coefficients in the sum $\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j=n^{2m+1}, \ m=1,2,…$

Consider a sum $$\sum_{k=0}^{n-1}\sum_{j=0}^{m}A_{j,m}(n-k)^jk^j$$ which returns an odd power $n^{2m+1}$ of $n$, for $\ m=0,1,2,...$ given fixed $A_{0,m}, \ A_{1,m}, \ ..., \ A_{m,m}$. The ...
1
vote
3answers
162 views

A problem with an edge labeling on the boolean lattices

Let $B_n$ be the boolean lattice of rank $n$. Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum, respectively. We identify the notion of edge with the notion of interval $[a,b]$ of cardinal $...
1
vote
1answer
139 views

Linear independence of +/- 1 strings/vectors

Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...
1
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0answers
87 views

Hook-content polynomial 2

Recently I have proven the following identity \begin{align} \sum_{\lambda\in \text{different hook of size d}} \frac{1}{d!} (-1)^{ht(\lambda)-1} \, \dim \lambda \, \prod_{\Box \in \lambda} \frac{1}{1-...
1
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0answers
78 views

Does the convex-hull of a set contain zero (II)?

Let $(\lambda_1, \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Suppose we have $k$ non-zero vectors $\alpha^{(j)} = (\alpha^{(j)}_1, \cdots , \alpha^{(j)}_d) \in \mathbb{Z}^d$, ...
1
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1answer
347 views

Nested De Bruijn Sequences

A binary De Bruijn sequence of index $n$ is a circular sequence $S=a_1 a_2 \dots a_{2^n},$ with $a_i \in \{0,1\},$ and such that each of the $2^n$ binary $n$-uples occurs exactly once in $S.$ Is ...
1
vote
1answer
182 views

Can this particular random matrix model be converted/related to any existing graph theory model?

Context: This a sequel to the question: Is the Erdős–Rényi giant component result applicable here? Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ ...
1
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0answers
345 views

Has anyone seen this version of ring toss (combinatorial object) before?

In reference to a question on work of Westzynthius and another question relating to Jacobsthal's function, I have formed a game which I immodestly call Paseman's Ring Toss. I hope that it has been ...
0
votes
1answer
97 views

Description of Linear Time Algorithm for TSP in Halin Graphs

I am looking for a description of the linear time algorithm for the TSP in Halin graphs, that was given in "G. Cornuejols, D. Naddef, and W.R. Pulleyblank. Halin graphs and the travelling ...
0
votes
1answer
210 views

Decomposition of a regular graph and connected subgraphs

I have asked almost same question earlier. I have been told that my question was poorly written, so I am trying to write it more clearly in this post. Also, this time I would be a little different in ...
0
votes
1answer
92 views

Walking “withouth gaps” through a set of sets

Let $X\neq \emptyset $ be a finite set and suppose that ${\cal C}$ is a set of subsets of $X$ with the following properties: $X\notin {\cal C}$, and for all $x,y\in X$ there is $A\in {\cal C}$ such ...
0
votes
0answers
164 views

Parity of number of partitions of $n!/6$ and $n!/2$

The parities of the number of partitions of $n!/6$ and $n!/2$ appear to be non-random initially, as follows — is there an explanation for this other than chance? With $p$ being the partition ...
0
votes
1answer
390 views

Asymptotic behaviour of Binomial Sum

I am interested in the behaviour of: $\gamma_k=\sum_{i=0}^{k} {n \choose i}$ as n becomes large and where $k$ could potentially be a function of $n$ rather than a constant. One line of attack I can ...
0
votes
0answers
117 views

Permutations $\pi\in S_n$ with $p_k+p_{\pi(k)}+1$ prime for all $k=1,\ldots,n$

As usual, let $S_n$ be the symmetric group of all permutations of $\{1,\ldots,n\}$. QUESTION: Is my following conjecture true? Conjecture. For any positive integer $n$, there is a permutation $\pi\...
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0answers
159 views

A combinatorial 0-1 matrix problem

Let $M \in \{0, 1\}^{n\times n}$. Given a constant integer $c \ge 2$, let the number of $1$s in each row be equal to $n/c$ (assuming $c$ is a divisor of $n$). Given a constant $\beta \in (0,1)$, we ...
-1
votes
2answers
249 views

Selection problem in a collection of non-empty sets

Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties? $a\in {\cal F} \implies |a|\geq 2$, $...
-1
votes
1answer
264 views

Odd & even permutations and unit fractions

One more motivated by recent questions of Zhi-Wei Sun. Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$. Is it true that, for every $n \ge 8$, there is at least one even permutation $\...
-2
votes
1answer
82 views

Hadwiger partitions where one block is always a singleton

Let $G=(V,E)$ be a simple, undirected graph. We call a partition ${\cal P}$ of a non-empty subset of $V$ a Hadwiger partition if every block (member of ${\cal P}$) is non-empty and connected, and if ...
-2
votes
1answer
79 views

Identifying two non-adjacent vertices and the effect on the Hadwiger number [closed]

Suppose $G$ is a finite simple graph. Let $h(G)$ denote the Hadwiger number of $G$; that is, the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$. What is an example of a graph $G_0=(V_0, ...
-4
votes
1answer
70 views

Regular graph such that $2$ distinct vertices have same neighborhood set [closed]

If $G=(V,E)$ is a simple, undirected graph and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in E\}$. Is there an integer $k>1$ and a connected $k$-regular graph $G=(V,E)$ such that there are $v\neq ...