All Questions
Tagged with co.combinatorics reference-request
334 questions with no upvoted or accepted answers
7
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259
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"Double convolution" with the Mobius function on a poset
Let $f$ and $g$ be arbitrary (say integer-valued) functions on some poset $P$, and say $\mu$ is the Mobius function of $P$. I'm studying a quantity that's a sort of "double convolution" of $f$ and $g$ ...
- 1,510
6
votes
0
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102
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The meet of two dominant permutations in weak order of $S_n$
A permutation is called dominant if its Lehmer code is a partition, or equivalently if it avoids the pattern $132$.
I can prove that given a permutation $v\in S_n$, there is a unique dominant ...
- 2,168
6
votes
0
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132
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Have the affine simplicial line arrangments been enumerated?
I am looking for a classification (or attempt at enumeration) of affine simplicial line arrangements.
A line arrangment is a family of straight lines in $\Bbb R^2$. It is simplicial if all regions are ...
- 13.6k
6
votes
0
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373
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Circle numbers on edges of a graph
Let $k$ vertices in a graph be given. Some pairs of vertices are connected by an edge, each edge is labeled either $\{1,2\}$, $\{1,3\}$, or $\{2,3\}$. We can circle some of the numbers on the edges. ...
- 277
6
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194
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"Cluster algebra" structure for finite distributive lattices
Let $P$ be an $n$-element poset and $J(P)$ the distributive lattice of its order ideals (i.e., the downwards-closed sets).
For each $I\in J(P)$ let $x_I \in \mathbb{R}^{n}$ be the indicator function ...
- 24.2k
6
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0
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365
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Is this just a numerical accident or what?
In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation
$$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m}
=\prod_{...
- 43.2k
6
votes
0
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381
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Reference request: colored Motzkin path interpretation of Catalan numbers
Recall that a Dyck path of length $2n$ is a lattice path in $\mathbb{Z}^2$ from $(0,0)$ to $(2n,0)$ consisting of $n$ up steps $U=(1,1)$ and $n$ down steps $D=(1,-1)$ which never goes below the $x$-...
- 24.2k
6
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0
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207
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Parameter independence of Stanley's "content formula". Why?
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 remarked following ...
- 43.2k
6
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214
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Looking for a combinatorial proof for an identity involving $q$-Catalan triangles
Let $C_n=\frac1{n+1}\binom{2n}n$ be the Catalan numbers. Following my earlier post on MO, one fine colleague asked me if there is a $q$-analogue of the identity formed by the so-called Shapiro's ...
- 43.2k
6
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89
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Maximal number of commuting functions of a finite set
Let $S$ be a finite set with $n$ elements and let $F_S$ denote the set of functions from $S$ to $S$. I wonder whether anything is known about the maximal cardinality of a commuting subset of $F_S$? A ...
- 23.2k
6
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189
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$X$-rays of permutations
Consider the set of permutations $\mathfrak{S}_n$, on $\{1,2,\dots,n\}$, and identify each element $\pi\in\mathfrak{S}_n$ with the corresponding permutation matrix.
There has been some study (e.g. ...
- 43.2k
6
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answers
105
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Long loops in critical random graphs
A simple calculation seems to show that the expected number $X_k$ of loops of length $k$ in a critical Erdös-Renyi random graph $G(n,n^{-1})$ is approximately given by
$$ \mathbb{E} X_k=\frac1{2k}{e^...
6
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186
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Combinatorial game similar to Sprouts
Is there a name for the following combinatorial game? Is there a solution which player has a winning strategy?
Basically this game is "Sprouts without midpoints". One starts with $n$ points in the ...
- 5,482
6
votes
0
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118
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Convex hull of all-ones principal submatrices
For a subset $S$ of $\{1,\ldots,n\}$,
let $\mathbf{1}_S\in\{0,1\}^n$ denote the indicator vector of $S$, with a $1$ on the $i$th coordinate iff $i\in S$. Let $\mathcal{X}$ denote the convex-hull of ...
- 617
6
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116
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Chromatic numbers for coloring-constrained graphs
I am interested in any and all articles about chromatic numbers applying to constrained colorings of a graph. For example, if a graph must be (properly) colored so that there is a 2-color path ...
- 91
6
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115
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Recursions which define polynomials?
Let $k$ be a positive integer and let
$$h(n,k,q)=\frac{1-(1+q^{k})q^{2k(n-1)+1}+q^{2}}{1-q^{2n-1}}h(n-1,k,q)-\frac{(1-q^{k(2n-3)})(1-q^{2k(n-1)})q^2}{(1-q^{2n-1})(1-q^{2n-3})}h(n-2,k,q)$$
with ...
- 5,622
6
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359
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Have topographs been studied before?
This is my first post on MO so I hope this question is suitable. I have quite a few definitions which I will need to state before my questions at the end of this post. Please let me know if anything ...
- 661
6
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380
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Large sets not containing arithmetic progressions of length 3 in intervals
Given a large enough natural number $N$, let $\Delta_N=\{A \subseteq [N, 2N]: A$ contains no arithmetic progressions of length $3 \},$ where for natural numbers $N<M$ we have $[N, M]=\{N, N+1, ..., ...
- 32.1k
6
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206
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What is the mobius function for the set of simplicial complexes on n vertices?
Consider the set of simplicial complexes on $n$ vertices, with partial ordering by containment. What is the Mobius function for this poset?
Are other combinatorial facts known about it (e.g. the ...
- 720
5
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answers
141
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If chromatic polynomials for two graphs agree, can I always find an edge such that the two deletion-contraction minors have same chromatic polynomial?
Suppose I have non-isomorphic graphs $G$ and $H$ (which have at least one edge), but such that their chromatic polynomials are the same. Can I then always find an edge $e$ in $G$ and $f$ in $H$ such ...
- 15.8k
5
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107
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Generalized Puiseux series for diagonal reflections of the curves $y = \frac{x}{(1-ax)(1-bx)^m}$
Reflection of the curve $y = f_m(x) = \frac{x}{(1-ax)(1-bx)^m}$ through the diagonal line $y=x$ in the $xy$-plane can be regarded as local compositional inversion of the curve $y=f_m(x)$. ($x,y,a,b$ ...
- 10.5k
5
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201
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Is this "trimming" of a supersolvable semimodular lattice known?
Let $L$ be a finite (upper) semimodular lattice. Recall that this means $L$ is graded and its rank function $\rho\colon L \to \mathbb{N}$ satisfies
$$ \rho(x) + \rho(y) \geq \rho(x\vee y)+\rho(x \...
- 24.2k
5
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190
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Yet, another generalization of Catalan determinants
The discussion on this page is motivated by Johann Cigler's MO question. My intention arose from a possible generalization of Cigler's matrix
$$A_{n,m}=\left( \binom{2m}{j-i+m}-\binom{2m}{m-i-j-1} \...
- 43.2k
5
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152
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Variant of the pancake problem
For two permutations $\pi,\tau \in S_n$, we say they are related by a prefix reversal if there exists $t$ such that $\tau(i) = \pi(i)$ for $i\ge t$ and $\tau(i) = \pi(t-i)$ for $i<t$. Similarly, we ...
- 3,499
5
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231
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Schröder and graphical logic?
I was actually surprised by a comment by John Baez over at the n-Category Cafe about his surprise that Ernst Schröder, a mathematician of whom he had known through Schröder's work on mathematical ...
- 10.5k
5
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325
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Self-dual simplicial complexes
A simplicial complex $K$ on a vertex set $[m] = \{1,...,m \}$ - where here we say vertex set to mean an ambient set of vertices on which $K$ is defined - is self-dual if it is equal to its Alexander ...
- 208
5
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132
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Asymptotics of Hilbert series for locally finite free graded-commutative algebras?
Let $A^\bullet$ be an $\mathbb N$-graded algebra over a field $k$, and let $d_A(n) = \dim A^n$ be the dimension of the $n$-th graded piece, so that $P^A(t) = \sum_n d_A(n) t^n$ is the Hilbert-Poincare ...
- 63.9k
5
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301
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The expressiveness of functions computable on trees
Motivation:
Let's define a function computable on a $k$-ary tree as a function composed with simpler computable functions defined at each node such that a function of this kind defined on a binary ...
- 3,871
5
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102
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An upper bound on the minimum number of vertices in a girth 5 graph of chromatic number $k$
Is there a known upper bound on the minimum number of vertices in a graph with girth 5 and chromatic number $k$? Could you also give references for this?
- 51
5
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150
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monomer-dimer tiling of a Young diagram
As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known.
Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
- 43.2k
5
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2k
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Jacobi's two-square theorem
Jacobi's theorem is: the number of ways of representing $N$ as a sum of two squares is $4(d_1(N)-d_3(N))$ where $d_i(N)$ is the number of divisors of $N$ that are of the form $4k+i$. I was wondering ...
- 221
5
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474
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Convex polytopes as "products" of lower dimensional polytopes of the same family
This MO-Q details the sense in which an associahedron is a product of lower dimensional associahedra, and this MSE-Q indicates the same is true for permutohedra.
Is there a reference which classifies ...
- 10.5k
5
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306
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Generalization of Sprague-Grundy Theorem
In my research on Combinatorial Game Theory, I used a certain theorem that is essentially a generalization of the Sprague-Grundy theorem. Because the result hinges too much on the work of others to be ...
- 1,129
5
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215
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Asymptotics of a Splitting Process
Consider $p(n)$ defined recursively by $p(1)=1$ and
$\displaystyle p(n)=\frac{1}{(n-1)^n}\sum_{i=1}^{n-1}\left\{\sum_{j=i}^{n-1}(-1)^{j-i}{n \choose j}{j\choose i}(n-j)^j(n-j-1)^{n-j}\right\}p(i)$.
...
- 667
5
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0
answers
227
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Number of times lead changes in a multi-candidate election (reference-request)
In a two candidate election where votes are distributed uniformly at random between the candidates, the probability that the lead changes when tallying the $i$-th vote is the same as the probability ...
- 51
4
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answers
91
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Reference for fact about flags of vexillary permutations
Vexillary permutations are an important family of permutations in Schubert calculus. There are several definitions, for example that they avoid the pattern 2143.
Recall the Lehmer code of a ...
- 1,989
4
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90
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Symmetric functions and pattern avoidance
It is known that the number of $k$-regular simple graphs with vertices labeled by $1,2,\dots,n$ can be expressed as the coefficient of $x_1^k \dots x_n^k$ in a symmetric function, which is
$$
\prod_{1\...
- 1,608
4
votes
0
answers
82
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Expansion of Schubert polynomials into standard elementary monomials
I have an explicit formula for expressing any Schubert polynomial in terms of standard elementary monomials that may or may not be cancelation-free. I haven't determined this yet, but it seems likely ...
- 2,168
4
votes
0
answers
89
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Software reference for combinatorial design
If one were to require quick and easy access to sizeable latin squares, room squares, Steiner systems, designs, balanced block designs... where to look, what software to use?
- 1,461
4
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0
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224
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A probabilistic proof of van der Waerden theorem
Is there an elementary proof of van der Waerden's theorem on arithmetic progressions using probabilistic methods?
- 32.1k
4
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0
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70
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A question about the existence of surjective contractions
A few years ago I was doing some research in origami, and was motivated to as the following questions:
Consider $\mathbb{R}^2$ with the Euclidean metric and Lebesgue measure. Does there exist a ...
- 384
4
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0
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97
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"Convolving" a general Catalan with classical Catalan
Consider what is sometimes known as generalized Catalan sequence
$$\mathcal{{\color{red}C}}_{a,b}:=\frac{2b+1}{a+b+1}\binom{2a}{a+b}.$$
Observe that $\mathcal{{\color{red}C}}_{n,0}$ reduces to the ...
- 43.2k
4
votes
0
answers
208
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Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
- 43.2k
4
votes
0
answers
181
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Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
4
votes
0
answers
135
views
Permutations avoiding a family of consecutive patterns
Let $B=\{1324,14325,154326,1654327,\ldots\}$ be the set of permutation patterns of the form $1(m-1)(m-2)\cdots 2m$ for $m\geq 4$. I'm interested in the set $\mathcal P$ of permutations that avoid all ...
- 135
4
votes
0
answers
89
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How to measure the optimality of the induced order by a median order of a tournament on a big subset
Median orders are great tools for dealing with a-priori unknown orientations of edges in tournaments, because they provide us with local properties on oriented edge density.
I've been wondering if ...
- 71
4
votes
0
answers
128
views
Inequality for $q$-binomials
Recall the constructions $[n]_q=\frac{1-q^n}{1-q}, [n]!_q=[1]_q[2]_q\cdots[n]_q$ with $[0]!_q:=1$ and the $q$-binomials (Gaussian polynomials)
$$\binom{n}k_q=\frac{[n]!_q}{[k]!_q[n-k]!_q}.$$
Given two ...
- 43.2k
4
votes
0
answers
234
views
To whom is the classification of atomic, modular finite lattices due?
Here lattice means a poset with meets and joins. A lattice is called atomic if every element is a join of atoms. There are a few different ways to define modular for finite lattices: one is that the ...
- 24.2k
4
votes
0
answers
163
views
An identity for Schur polynomials
Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as
$$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...
- 43.2k
4
votes
0
answers
186
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A problem in the spirit of P. Borwein's polynomials
A well-known conjecture (now a theorem) of P. Borwein (see Wang and Krattenthaler - An asymptotic approach to Borwein-type sign pattern theorems) states:
For all positive integers $n$, the sign ...
- 43.2k