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.
1,457
questions
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Smith Normal Form of powers of a matrix
What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...
17
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8
answers
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Cyclotomic polynomials in combinatorics
I am searching for a combinatorial significance of cyclotomic polynomials. The only examples I got are a paper by Neville Robbins http://www.emis.de/journals/INTEGERS/papers/a6/a6.pdf and two recent ...
17
votes
1
answer
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Reference request: a conjecture of Rota on positive functions of a random variable
Rota and Shen's On the Combinatorics of Cumulants ends with a conjecture which I'll restate as follows:
Let $p \in \mathbb{R}[x_1, x_2, ...]$ be a polynomial such that, for any sequence $X_1, X_2, ...
17
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11
answers
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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 ...
17
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2
answers
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Number of isomorphism types of finite groups
Are there some good asymptotic estimations for the number $F(n)$ of non-isomorphic finite groups of size smaller than $n$?
16
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4
answers
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Exact formulas for the partition function?
I am curious, what kind of exact formulas exist for the partition function $p(n)$?
I seem to remember an exact formula along the lines $p(n) = \sum_k f(n, k)$, where $f(n, k)$ was some extremely ...
16
votes
3
answers
701
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Can we realize a graph as the skeleton of a polytope that has the same symmetries?
Given a graph $G$, a realization of $G$ as a polytope is a convex polytope $P\subseteq \Bbb R^n$ with $G$ as its 1-skeleton.
A realization $P\subseteq \Bbb R^n$ is said to realize the symmetries of $...
16
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0
answers
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A Product Related to Unrestricted Partitions
Start with the product for unrestricted partitions:
$(1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)...$
Now replace some of the plus signs with minus signs and expand the product into a series. Is it ...
15
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4
answers
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Ordinary Generating Function for Bell Numbers
In the OEIS entry for Bell numbers, there appears a generating function
$$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$
However, I could not locate any proof of ...
15
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3
answers
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Integration of a function over 7-sphere
Suppose we have $x_1^2 + y_1^2 + x_2^2 + y_2^2 + x_3^2 + y_3^2 + x_4^2 + y_4^2 = 1$ and we define $z_j = x_j + iy_j$, where $j = 1,\,2,\,3,\,4$.
The problem is finding or approximating the ...
14
votes
4
answers
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Number of closed walks on an $n$-cube
Is there a known formula for the number of closed walks of length (exactly) $r$ on the $n$-cube? If not, what are the best known upper and lower bounds?
[Edit] Note: the walk can repeat vertices.
14
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5
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A general formula for the number of conjugacy classes of $\mathbb{S}_n \times \mathbb{S}_n$ acted on by $ \mathbb{S}_n$
$\def\S{\mathbb{S}}$ Dear all,
So I have $\S_n$ acting on $\S_n \times \S_n$ via conjugacy. That is:
for $g \in \S_n, (x,y) \in \S_n \times \S_n$: $g(x,y) = (gxg^{-1},gyg^{-1}).$
Is there a general ...
13
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1
answer
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Irreducibility of Schur polynomials
A natural question covering both this and this question would be
Let $n>2$. Describe Young diagrams $\lambda$ with at most $n$ nonempty rows (or equivalently non-increasing sequences $\lambda=(\...
13
votes
3
answers
851
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A strengthening of Frankl's union-closed conjecture?
Frankl's union-closed conjecture states that if $F$ is a finite union-closed family of sets (i.e. a family that is closed under taking unions), then there must be an element that belongs to at least ...
12
votes
1
answer
842
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The dance marathon problem
In his book, "The Strange Logic of Random Graphs", Joel Spencer describes the "Dance Marathon" problem:
Imagine $n$ couples at a Dance Marathon. Each dance each couple remains ...
12
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3
answers
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Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
The question is stated in the title of this post.
It is easy to see that, if $z$ is a multiple root of $p_n(z):=1+z+z^2-z^n$, then $(n-2)z^2+(n-1)z+n=0$, so that we can successively express $z^2,\dots,...
12
votes
0
answers
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Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
Definition / Question
Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where
$0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$...
12
votes
1
answer
483
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Probability of a graph procedure
We are going to build $K_n$ one edge at a time. Begin with the empty graph on $n$ vertices. Take a random permutation of the edges of $K_n$ and, one at a time, place the edges onto the graph (so, ...
12
votes
2
answers
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Combinatorics of the Stasheff polytopes
First a little background for those unaware. The Stasheff polytopes (or associahedra) are certain convex polytopes that arise in the theory of $A_\infty$-algebras. There is one polytope for each $n\...
11
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11
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Lattices on classical combinatorial families
I am asking for examples of lattices defined on classical combinatorial families, such as Permutations, Catalan objects, set partitions or integer partitions, graphs.
I am mosty interested in lattices ...
11
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2
answers
995
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Densest Graphs with Unique Perfect Matching
Given a graph $G$ with $n$ vertices, that has a perfect matching $M$, what is the maximal number of edges that $G$ can have without contradicting the uniqueness of $M$?
Are examples of such extremal ...
11
votes
1
answer
673
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2-adic Logarithm and Resistance of n-dimensional Cube
Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is
$$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=0}^{n-1}\frac1{{n-1\choose k}}.$$
The ...
11
votes
1
answer
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Catalan determinants in search of a proof: Part II
This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.
I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof.
...
10
votes
1
answer
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How can we find n points on a plane so that as many pairs of points as possible have the same distance?
There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
9
votes
1
answer
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The line graphs of complete graphs and Cayley graphs
Let $n>3$ be an odd integer and let $K_n$ denote the complete graph on $n$ vertices.
For which integers $n$ the line graph $L(K_n)$ is a Cayley graph?
For even $n$, it follows from a result of ...
9
votes
2
answers
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Sperner's lemma and Tucker's lemma
In their article "A Borsuk-Ulam Equivalent that Directly Implies Sperner's Lemma" (American Mathematical Monthly, April 2013), Nyman and Su write "[W]e are unaware of a direct proof that Tucker's ...
9
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0
answers
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Factorisation of a polynomial from the Boolean algebra
Let $B_n$ denote the Boolean algebra of a set with $n \geq 2$ elements and $C_n$ the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else, where $i,j\in B_n$.
Let $M_n:=C_n+C_n^T$ and $...
8
votes
3
answers
703
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Combinatorial interpretation of series reversion coefficients
In a recent paper studying some generalizations of Stirling numbers, my coauthors and I needed the following result:
If $f(x)=\sum_{n \geq 1} a_n x^n/n!$ is a power series with $a_1 \neq 0$, and $g(x)...
7
votes
0
answers
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How distributive are the fake Laver tables?
The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$.
Let's now replace the Laver table $A_{n}$ with ...
7
votes
2
answers
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Limit associated with complementary sequences
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
6
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2
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How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph?
I have a question about the combinatorial Laplacian $\Delta$ which is defined by
$$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$
where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a ...
6
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3
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An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
What are the roles that the classic number arrays-- Eulerian, Narayana--play in the application of totally non-negative Grassmannians, or amplituhedrons, to string / twistor scattering theory?
(This ...
6
votes
1
answer
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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 ...
6
votes
1
answer
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hooks and contents: Part I
For a cell $\square$ in the Young diagram of a partition $\lambda$, let $h_{\square}$ and $c_{\square}$ denote the hook length and content of $\square$, respectively.
R Stanley proved the following ...
5
votes
1
answer
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differential operator power coefficients
Let $(F(x)\frac{d}{dx})^n=\sum_{i=1}^n H_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H_{n,i}(y_0, y_1, \ldots , y_n)$. What is known about it?
4
votes
1
answer
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Is the top interval of a finite distributive lattice, a boolean lattice?
Let $(L,\wedge,\vee)$ be a finite distributive lattice, and let $1$ its greatest element.
An element $a \in L$ is called maximal if $a \le a' < 1$ implies $a = a'$.
Let $b$ be the meet of all the ...
4
votes
3
answers
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Minimal data required to determine a convex polytope
Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...
4
votes
2
answers
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Estimate on sum of squares of multinomial coefficients
I am interested in approximating the sum of the squares of the multinomial coefficients, i.e.
$a_\ell^p := \sum_{k_0+\ldots+k_p = \ell} (\frac{\ell!}{k_0! \ldots k_p!})^2$
or more general,
$a_\...
4
votes
1
answer
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Generating functions for Hankel determinants of Catalan numbers
The Hankel determinants of the Catalan numbers are well known and can be written as
$d(k,n)= \det \left( C_{k + i + j} \right)_{i,j = 0}^{n - 1}=\prod_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$...
3
votes
1
answer
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An identity for polynomials over partitions
Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its conjugate partition $\lambda'$. Denote by $\lambda'...
3
votes
1
answer
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The number of pairings between multisets
Given two multisets $A$ and $B$ of the same finite cardinality $n$, how many ways are there of pairing the two sets together?
If both sets consist of distinct elements, the answer is $n!$: there are $...
3
votes
0
answers
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A conjectural lower bound for $|\{\sum_{k=1}^nka_k:\ a_1,\ldots,a_n\ \text{are distinct elements of }\ A\}|$
Motivated by Question 315568 of mine, I'm interested in the set
$$S(n):=\bigg\{\sum_{k=1}^n k\pi(k):\ \pi\in S_n\bigg\}.$$
It is easy to see that
$$S(1)=\{1\},\ S(2)=\{4,5\}\ \text{and}\ S(3)=\{10,...
2
votes
2
answers
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Number of bounded Dyck paths with negative length as Hankel determinants
This is a continuation of my post Number of bounded Dyck paths with "negative length".
Let $C_{n}^{(2k+1)}$ be the number of Dyck paths of semilength $n$ bounded by $2k+1.$ They satisfy a ...
1
vote
0
answers
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Classical and free cumulants, symmetric functions, and inverses (references), related to associahedra, parking functions, noncrossing partitions
Looking for references for one or more of the following four sets of partition polynomials 1a) through 4a), particularly those which present geometric / topological combinatorial interpretations.
...
1
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0
answers
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On the number of Eulerian orderings
This post is a sequel of Eulerian ordering of the integers modulo n.
Let us recall the definition of an Eulerian ordering:
Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$....
1
vote
1
answer
223
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Will this greedy algorithm always work?
Let p(n) be the number of unrestricted partitions of n. p(0) is taken to be 1.
Let set 1 and set 2 be two empty sets.
Here's an algorithm. Put p(n) into set 1. On each successive step, k=1,2,3,..., ...
29
votes
3
answers
4k
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Roots of permutations
Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $...
111
votes
7
answers
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Is the set $ AA+A $ always at least as large as $ A+A $?
Let $A$ be a finite set of real numbers. Is it always the case that $|AA+A| \geq |A+A|$?
My first instinct is that this is obviously true, and there is a one-line proof which I am foolishly ...
101
votes
3
answers
6k
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Why do combinatorial abstractions of geometric objects behave so well?
This question is inspired by a talk of June Huh from the recent "Current Developments in Mathematics" conference.
Here are two examples of the kind of combinatorial abstractions of geometric ...
70
votes
7
answers
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Identifying poisoned wines
The standard version of this puzzle is as follows: you have $1000$ bottles of wine, one of which is poisoned. You also have a supply of rats (say). You want to determine which bottle is the poisoned ...