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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14 views

Computing row-sum scaled unsigned Stirling nos. of the 1st kind

As the title suggests, I would like to compute probability vectors with elements proportional to (unsigned) Stirling numbers of the first kind by row. For easy reference, here is the Wiki page. For ...
1
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0answers
12 views

Semigroup nilpotents and compostional inversion

The integer coefficients of a general partition formula for the compositional inverse of a function are a refined version of the coefficients of the generating series for the number of nilpotents in a ...
3
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0answers
33 views

Probability that a sum of intependent random variables hits a point

Let $X_1,\ldots,X_n$ be independent random elements of a normed space $X$. Suppose that $\sup_{x\in X}\mathbb{P}(X_i=x)=p_i$. What is the best known upper bound for $$\sup_{x\in X} ...
4
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2answers
90 views

Are sums of 0-1 Pareto efficient vectors Pareto efficient?

Does there exist an $m \times n$ matrix $A$ and a vector $x \in \mathbb{R}^m$ such that: The entries of $A$ are $\in \{0, 1\}$. For all pairs of columns $u, v$ of $A$ the entries of $u - v$ are ...
3
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0answers
113 views

What is God's number for the WrapSlide puzzle?

WrapSlide is a slide-puzzle (reminding of Rubik's Cube) consisting of a 6x6 grid of coloured tiles which are separated into four quadrants of 3x3 tiles. When it is unmixed all the tiles in a quadrant ...
3
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1answer
104 views

When are all sums of the elements of a set different?

Consider a set $S = \{x_1, \dots, x_n\} \subset \mathbb{Q}\setminus\{0\}$ and assume that for any $I, J \subset [n]$ with $I \neq J$ we have that \begin{equation} \sum_{i \in I} x_i \neq \sum_{j \in ...
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47 views

K-Permutations with forbidden numbers [on hold]

This question has some references to programming and not as many mathematical terms as you might like, but I think it's more appropriate in a mathematics forum. Introduction (Skip if you are ...
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70 views

Combinatorial Proof Problem [on hold]

I'm having trouble solving this because I'm only familiar with algebraic proofs instead of combinatorial. $$\binom{3n}{3}=n^3+6n\binom{n}{2}+3\binom{n}{3},\quad\text{for }n\ge3.$$
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57 views

A problem on chains of squares — can one find an easy combinatorial proof?

Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection $$ A = \left\{ \left[ \frac{i}{n},\frac{i + 1}{n} \right] \times ...
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51 views

Number of maximal chains in Bruhat order

Is there a formula for the number of maximal chains between two permutation in the (strong) Bruhat order?
3
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1answer
142 views

A number array related to colored necklaces and the primes

I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
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88 views

What is known about the chromatic number for minimum-distance graphs in higher dimensions?

For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of ...
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0answers
60 views

Fundamental invariants for root subsystems

Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$. Now suppose $\Psi$ is ...
7
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146 views

On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
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2answers
332 views
+50

Separating points in the plane II

Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all ...
5
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1answer
382 views

Is fourier analysis necessary to prove this?

I have a couple of inequalities that I want to prove. The proof is easy using fourier analysis but I am wondering whether there is a proof that does not use fourier analysis. 1) For any $c, s > ...
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43 views

quadratic knacksack problem, state of art [closed]

What is the current status to quadratic knacksack problem? Say, how many variables can the state of art solver handle? Thank you.
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38 views

Combining binary numbers [closed]

Assume you have n n-digit binary numbers, for example: ...
0
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1answer
89 views

Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...
1
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1answer
149 views

A generalisation of Narayana-like numbers (walks on the 2D lattice)

I apologize in advance if this question has a trivial answer. I am pretty sure this kind of problem was already studied and I am mostly asking for good references. Given integers $0 < k \le n+1,$ ...
1
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1answer
106 views

Ordinary or Rational Generating Function for Associated Stirling Numbers $b(n,k)$

I am trying to identify or find the ordinary or rational generating function (not the exponential generating function) for the Associated Stirling numbers of the Second kind, denoted ...
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2answers
234 views

Does a classification of simultaneous conjugacy classes in a product of symmetric groups exist?

Let the symmetric group $S_n$ on $n$ letters act on $S_n^d=S_n\times\cdots\times S_n$ by simultaneous conjugation, i.e. $\pi\in S_n$ acts on $(\sigma_1,\ldots,\sigma_d)\in S_n^d$ by ...
3
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1answer
62 views

Are all marked order polytopes normal?

Richard Stanley showed that order polytopes have a unimodlar ttriangulation. In particular, this implies that they are integrally closed/normal. One can generalize order polytopes to marked order ...
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1answer
154 views

Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$. Let $k\ge0$ be a nonnegative integer. If we add another factorial ...
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352 views

What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$

Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...
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0answers
100 views

current status of combinatorial optimization solvers [closed]

What is the current status of the solvers in combinatorial optimization? For example, what is the "usual feasible size" for a traveller sale's man problem, say, how many nodes and edges are "usually ...
2
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2answers
118 views

Choosing subsets to cover larger sets

Consider the set $S=\{1,2,\ldots, n\}$, and let $a<b<n$. What is the minimum number $f(a,b)$ such that there exist $f(a,b)$ subsets of $S$ of size $a$ for which any subset of $S$ of size $b$ ...
3
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1answer
45 views

Average vertex degree in finite Delaunay triangulations in high dimensions

In $\mathbb{R}^2$ it's known that with a "random" point configuration, the average degree of a vertex in its Delaunay triangulation is 6. Does anyone know of a similar result in higher dimension? I ...
2
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2answers
83 views

Probability of a contiguous sub-sequence with different elements

Let $a$ and $b$ be two positive integers, and say $b\gg a$. Let $S$ be a random sequence with $ab$ elements, whose entries are all integers from $1$ to $a$, such that each number from $1$ to $a$ ...
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1answer
306 views

Coloring of the plane

I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color ...
6
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1answer
177 views

Some polytopes in $\mathbb R^n$ whose vertices have coordinates 1, -1 or 0

Let $n$ and $k$ be positive integers with $k\leq n$. Let $P(n,k)$ be the convex hull in $\mathbb R^n$ of the $2^k {n \choose k}$ vectors whose exactly $k$ coordinates belong to $\{\pm 1\}$ all the ...
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1answer
92 views

Evaluation of the multiple integral [closed]

Would you give me any suggestions or comments on evaluating the following $n$-dimensional integral? $$ \int_{[0,t]^n} h(x) dx $$ where $ x=(x_1 ,x_2 , \cdots, x_n ), h(x)= \prod_{k=1}^n min( ...
2
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0answers
43 views

Computing basis of a lower set given basis of complementary upper set

In a poset $P$, $U\subseteq P$ is an upper set when for all $x\in U$, we have $y\ge x$ implies $y\in U$. Any subset of $P$ generates an upper set, and the basis of an upper set $U$ is the smallest ...
2
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0answers
76 views

Counting strings with alternating letters with generating functions

It is a classical problem that of finding the generating function (GF) of the number of strings with length $n$ having $m$ different letters (basically, the problem reduces to that of writing the ...
3
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74 views

Counting Problems where Labeled is Known but Unlabeled is Not

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. There are many nice proofs of this compact formula. To contrast, counting unlabeled trees is considerably ...
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3answers
93 views

Is there a formula for the number of labeled forests with $k$ components on $n$ vertices?

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. My question is: Is there a generalization of this formula for forests? Let $f_{n,k}$ denote the number of ...
5
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1answer
161 views

How to prove an elementary functional equation for polylogarithms?

Let $Li_s(z)$ denote the usual polylogarithm. The elementary functional equation $$Li_{-n}(z)=(-1)^{n-1}Li_{-n}(1/z)$$ holds for $n\geq 1$. I remember only that the proof used some reproducing ...
4
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1answer
209 views

Generalized Cauchy-Binet sum over a fixed subset of indices

I originally posted this on math.stackexchange, but it quickly got buried. I removed it not too long after, thinking of rewriting it for MO, but I didn’t have a chance to post it until now. Apologies ...
4
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1answer
488 views

An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

$$Prelude \;\;on \;\;Two\;\; Threads$$ Last month the workshop New Geometric Structures in Scatering Amplitudes was hosted by CMI with the following partial overview: Recently, remarkable ...
2
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0answers
46 views

Counting rooted labelled trees with fixed height AND fixed # of vertices

Consider the set of rooted labeled trees in which each node may have up to $d$ children. It is well known that the number of such trees with $\leq v$ vertices is (approximately) $(ed)^v$ and that the ...
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74 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 ...
2
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1answer
92 views

enumeration of connected blocks in finite size square

Given a square of size n by m, how many ways could we choose sites, such that all the sites are connected? By "connected" we mean "connected" by adjacent sites. We will illustrate by example, say, we ...
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100 views

lower bound for difference between max cut and min cut

Let $G=(V,E)$ be a graph on $n$ vertices with edge weights $w=(w_e)_{e\in E}$. Let $M^+$ and $M^-$ be the maximum and the minimum cut values, i.e. ...
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1answer
138 views

How to understand a rooting of a dessin d'enfant?

As I understand it, rooted maps on surfaces were first introduced in enumerative combinatorics because they are easier to count than unrooted maps, which can have non-trivial symmetries. A map is a ...
4
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2answers
135 views

open question on intersecting rectangles - reference request

In the now-inactive (and very nice) "Tricki" website that describes various proof techniques, there is an article (written I think by Timothy Gowers) that describes the following Ramsey-type problem: ...
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137 views

Dissection of a polygon into convex polygons

Problem: for a fixed integer $m\geqslant 3$ find all $n$ such that no $n$-gon can be dissected into convex $m$-gons. I would be very grateful for any information on this problem. Remark 1. There ...
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2answers
139 views

Upper bounding number of integer partition by binomial coefficient

Let $n,p \in \mathbb N$ and consider the integer partition of $$\lfloor \frac{n(p-1)}{2} \rfloor$$ into $p$ or less parts, each of which is less or equal to $n-1$. Can the number of partitions be ...
3
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2answers
178 views

Upper bound for different sums having the same value

Let $\Lambda :=\{\lambda_1, \dots, \lambda_n\}$ be a set of $n$ distinct real numbers. For a given $p \in \mathbb N$, consider further the set $$I_p := \{ \{i_1, i_2, \dots, i_p\} : i_j \in \{1, ...
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1answer
132 views

How many maximal triangulations of a rectangle?

I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow. Let $L_{m,n}\subset\mathbb R^2$ be a rectangle given by $[0,m]×[0,n]$ with ...
5
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54 views

Independent domination number for grid graphs

Let $G_{n,m}$ be the $n \times m$ grid graph, i.e. $G= P_n \Box P_m$, and $T_{n,m}$ the $n\times m$ torus grid graph, i.e. $G= C_n \Box C_m$, where $P_n$ and $C_n$ indicate the path graph of length ...