All Questions
Tagged with co.combinatorics pr.probability
802 questions
12
votes
3
answers
1k
views
A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper?
In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $p$. From what I ...
- 470
7
votes
1
answer
876
views
What is the six positive real number for a dice producing a highest chance?
Say there is a dice with six faces, each face has a positive real number different from others. There is a chessman on the origin of the number axis. In each trial, the dice rolls infinite times. ...
- 143
7
votes
0
answers
171
views
What is known about the distribution of lengths of the cycle you get by adding an edge to a uniform spanning tree?
Let $G$ be a finite, connected graph. Let $T$ be a uniform spanning tree, and let $e$ be a uniformly random edge not in $T$. When we add $e$ to $T$, we get a subgraph with a unique cycle, $C$. I am ...
- 1,462
2
votes
1
answer
607
views
Component size distribution in small Erdos-Renyi networks
I'm looking at $\mathcal{G}(n,p)$ (I'll call these Erdos-Renyi networks) where $n$ is, say, at most 10.
I would like to know the probability a random node is in a component of size $m$.
It's ...
- 121
3
votes
1
answer
167
views
Lower bound on the sum of pmf squared of a hypergeometric distribution
I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant ...
- 33
5
votes
1
answer
281
views
Transfer-impedance matrix for edge correlations in random spanning tree
Suppose $G$ is a (weighted) connected graph and
let $T$ denote a random spanning tree of $G$,
chosen uniformly (or respecting the edge weights).
It is known that for any distinct edges $e, f$
$$\...
- 2,175
2
votes
1
answer
188
views
Colored balls and bins -- asymptotic behavior
Suppose I have $N$ bins and a set of balls with $m$ different colors, where there are $n_i$ balls of color $i$. I also have values $0 < p_i \leq 1$ for all colors $i$. I throw all $\sum_i n_i$ ...
- 4,049
4
votes
1
answer
648
views
What is the probability for a Binomial to be greater than other?
Let $X = B(n, p)$ and $Y = B(k, q)$ be two random variables with binomial distribution and, let $s$ be a positive integer. Assume that $n > k+s$ and $np \geq kq+s$.
What is the probability for $X$ ...
4
votes
1
answer
239
views
$Pr(A>B)$, where $A$ and $B$ are sum of Bernoullies
Let $X= x_1 + x_2 + \ldots + x_m$, $Y=y_1 + y_2 + y_3 + \ldots + y_n$, and $Y' = y'_1 + y'_2 + \ldots + y'_n$, where
Each $x_i$ is a Bernoulli variable which takes value $1$ with probability $p_i>...
- 169
2
votes
0
answers
78
views
Width of symmetric groups
MSE crosspost
For any (finite) group $G$ its length $l(G)$ is the length of maximal chain of proper subgroups (it's known and pretty widely used invariant). But we can also define width function $w_G(...
- 4,600
2
votes
1
answer
90
views
Generalization: (The "number" of) smaller sized clusters in large random binary matrices follow a descending order. Why?
This is a sequel to the question: Why is number of single cell clusters always greatest in a random matrix?
In their answer, @Aaron Meyerowitz came up with a nice strategy to prove why the number of ...
user125648
3
votes
1
answer
184
views
Why is number of single cell clusters always greatest in a random matrix?
Consider a large $N\times N$ square lattice, where each cell has a probability $p$ of being "occupied" (let's call denote them as "black") and a probability $1-p$ of being empty (let's denote them as "...
user125648
15
votes
1
answer
1k
views
Math journal publishing work related to combinatorics, probability, counting problems etc.?
I'm a high school student. My peer and I have done some work on the Ballot Theorem counting problem and Catalan Numbers. We have come up with a new proof to the Ballot Theorem and we demonstrate the ...
0
votes
0
answers
119
views
the enumeration of 2 dimensional lattice walks with fixed number steps and largest distance from the end point ti the origin
There is actually an one dimensional version of this problem. For each step of the lattice walk, we can move either east for one unit or west for one unit. The problem is that given a fixed $n$ steps ...
- 167
5
votes
1
answer
340
views
Show a sequence of sums involving Catalan Numbers converges
Let $C_n$ be the $n$-th Catalan Number and let $\mathcal{O}_{s,j} = {{2s-j-1}\choose{j}} C_{s-j}^2$. Then we want to consider $\mathcal{E}_s = \sum_{j=0}^{s-1} (-1)^j\mathcal{O}_{s,j}$. We want to ...
- 313
3
votes
1
answer
228
views
Density of a somewhat random set
The density of a set
$X\subseteq\omega$ refers to:
$\limsup\limits_{n\rightarrow\infty}\dfrac{C\cap n}{n}$.
Given a set of positive integers
$F= \{m_0<\cdots<m_{k-1}\}$,
let $C\subseteq \omega$...
- 909
1
vote
1
answer
114
views
Probability for a group of stones to live on an infinite Go board
Suppose on an infinite two dimensional Go board the tengen is occupied by a black stone, and every other grid point is occupied by a black stone, or a white stone, or nothing, with probability 1/3 ...
- 5,169
3
votes
2
answers
503
views
Lower bound for the probability of binomial variable to be less than her expectation
Let $X$ a binomial variable of parameter $(N,p)$, with $0<p<0.5$
I would like to lower bound $\mathbb{P}\left(X <Np \right)$ by a constant ($\frac{1}{5}$ seems true and is enough for me).
...
- 215
7
votes
3
answers
790
views
Expected cardinality of a randomly chosen element of the family of subsets of $\{1,\ldots,n\}$ with at most $k$-elements
Assume that $1\le k \le n$ and let $\mathscr{Z}$ be the family of all subsets of $\{1,\ldots,n\}$ with at most $k$ elements. Pick a random element $X$ of $\mathscr{Z}$ (we consider the probablity ...
- 1,658
7
votes
1
answer
390
views
Combinatorial/probabilistic statements having $F_{\text{un}}$/$F_q$ geometric interpetation
$\newcommand{\Fun}{F_\text{un}}$There was lots of "Fun with $\Fun$" (field with one element) in recent years.
One of the points is that it provides bridge between geometrical and ...
- 24.9k
11
votes
3
answers
243
views
How many operad structures are there on the symmetric sequence of simplices / finitely-supported probability measures?
Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by ...
- 63.9k
0
votes
0
answers
72
views
Generating function for number of r-disjoint subsets each of size k
Fix $n, k$. Let
$$
C^{n,k}_r =\frac{1}{r!} \binom{n}{\underbrace{k, \ldots, k}_{\text{r times}}, n-rk} = \frac{n!}{r!(k!)^r(n - kr)!}
$$
be the number of ways to form $r$ disjoint subsets each of ...
- 101
2
votes
1
answer
508
views
Proof and interpretation of the following percolation theory result for $n\times n$ square grid
While I was discussing this question with @JamesMartin, he mentioned a result here that:
In a $n\times n$ finite square grid, if $p\geq p_c+\epsilon$, such
that $\epsilon>0$ and $p_c$ is the ...
user123818
2
votes
1
answer
299
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$ ...
user123818
-1
votes
1
answer
76
views
Transforming random variables for having good property?
For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that
\begin{align}
\Omega&\triangleq \{(x,y): A(x,y)=1\},\\
\Lambda&\triangleq \{x: B(x)=1\}.
\end{...
- 287
2
votes
2
answers
357
views
Is the Erdős–Rényi giant component result applicable here?
Consider a matrix whose elements are independently assigned a value $1$ with probability $p$ and a value $0$ with probability $1-p$.
Define a cluster of cells as a maximal connected component in the ...
- 41
2
votes
1
answer
132
views
Independent decomposition of coordinate distribution
Let $\mathbf{x}$ be a random Gaussian vector in $\mathbb{R}^n$, i.e. $\mathbf{x}\sim\mathcal{N}(\mathbf{0},\mathbf{I}_n)$. Then for any fixed unit vector $\mathbf{u}$, one has $\mathbf{u}\mathbf{u}^\...
- 515
3
votes
0
answers
116
views
Trace of Symmetric matrices in fixed rank
I am solving some problem related to symmetric matrices over a finite field $\mathbb{F}_q$ and I am stuck at the following problem:
For every $a\in\mathbb{F}_q $, let $S_a(t,m)$ be the set of all $m\...
- 179
3
votes
0
answers
178
views
Partitioning the coupons collected in the classical coupon collector's problem
Suppose that there is an urn containing $n$ different coupons, from which $m$ coupons are being collected, equally likely, with replacement.
Let $C(m)$ be the whole set of the $m$ collected coupons. ...
- 1,677
4
votes
2
answers
261
views
Probability question about random shuffling of piles of rocks
I have $k$ piles of rocks placed on a circle so that every pile has exactly two neighboring piles. We know that initially the piles have $x_1,\dots,x_k$ rocks in each respectively. A monkey plays the ...
- 41
8
votes
2
answers
240
views
Mixed moments for the birthday problem
Let $X_1,X_2,\dots$ be iid draws from the uniform distribution on $\{1,2,...,m\}$, and let the random variable $N$ be the minimum $j$ such that $X_j = X_i$ for some $i<j$.
I'm aware that the ...
- 19.7k
4
votes
2
answers
432
views
How to prove the sum of n squared binomial probabilities does not increase as n increases
Let $F\left( n \right) = \sum\limits_{k = 0}^n {{{\left( {C_n^k{p^k}{{\left( {1 - p} \right)}^{n - k}}} \right)}^2}} $, prove $F\left( n \right) \ge F\left( {n + 1} \right)$.
UPDATE: More general, ...
- 43
11
votes
1
answer
284
views
Asymptotic distribution of $\lambda_1$ under the $z$-measure for partitions
The following question about $z$-measures on Young diagrams came up in some ongoing work with Ofir Gorodetsky. I recall the background and then state our question below in the box.
For parameters $z$ ...
- 2,151
4
votes
0
answers
111
views
Set version of ramsey type problem
For two sets of numbers $A,B$, write $A<B$ iff $\max A<\min B$.
For a sequence of integers $a_0,\cdots,a_{n-1}>0$,
let $Prop(a_0,\cdots,a_{n-1})$ denote the following proposition:
Given $n$ ...
- 909
6
votes
1
answer
349
views
Ramsey type theorem
Let $\mathcal{P}(\{0,\dotsc,7\})$ denote the power set of $\{0,\dotsc,7\}$.
Is the following true?
For any function $f: \mathcal{P}(\{0,\dotsc,7\})\rightarrow\{0,1\}$ there exists $0\leq k\leq 3$ ...
- 909
11
votes
1
answer
867
views
Simulate coin tossing by die tossing
On the one hand we toss $n$ times a fair coin, and we sum the outcomes (+1 for heads, -1 for tails). Let $f:\mathbb{N}\to\mathbb{R}$ describe the probability distribution of the outcome.
On the other ...
- 338
1
vote
1
answer
188
views
KPZ relation $\chi = 2 \xi -1$ in a random geometric graph
If I have $n$ points uniformly distributed on the surface of a torus, and form a graph by adding an edge between pairs whenever they are within a unit distance (induced by the Euclidean metric), I ...
- 640
6
votes
1
answer
224
views
A Markov consensus
Consider the following process. You start with $n$ nodes in different colors $c=c1,c2,...$ (representing an opinion). Say, $n=5, c=1,2,3,4,5$. Now each node checks which colors have weak majority (...
- 4,829
4
votes
0
answers
355
views
Distribution of min/max row sum of matrix with i.i.d. uniform random variables
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $1$.
all elements in upper triangle (excluding the diagonal) are i.i.d. uniform random variables ...
- 272
2
votes
0
answers
59
views
Min/max row-sum distribution of a symmetric matrix of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix such that
all diagonal elements are all fixed as $0$.
randomly select $k$ distinct cells in the upper triangle (excluding the diagonal), and then ...
- 272
7
votes
1
answer
880
views
Bound for largest eigenvalue of symmetric matrices of uniform random variables over $[0,1]$ and fixed $1$s along diagonal and scattered $1$s
Given a $n\times n$ symmetric random matrix whose diagonal elements are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding ...
- 272
1
vote
1
answer
154
views
Is the Krawtchouk ensemble a determinantal process?
The Krawtchouk ensemble is defined by a weight: $w(x) = \binom{K}{x}p^x q^{K-x} $ and in fact it comes from a conditioned random walk on $\mathbb{Z}^N$. It is a probability measure on the set $\{ 0, ...
- 22.8k
3
votes
2
answers
1k
views
Proof of identity involving Stirling numbers of the second kind
While computing conditional expectations of certain functionals of a Poisson white noise field (details are long and probably irrelevant), I've stumbled upon the need to use the following identity ...
- 393
2
votes
1
answer
266
views
A question about finite free convolution
For any square matrix $Y$ let $\chi_x(Y) = det(xI -Y)$ denote its characteristic polynomial.
Say $A$ and $B$ are two $n-$dimensional symmetric matrices with constant row sums $a$ and $b$. Lets ...
- 2,246
5
votes
0
answers
352
views
0-1 matrix combinatorial problem
Let $M \in \{0,1\}^{n \times n}$ and let $r_i$ be its $i$-th row. Given constant $p \in (0,1/2]$, let the number of $1$s in each row be at least $p\,n$. Given constant $c \in (0,1)$, what is the ...
- 1,677
3
votes
2
answers
228
views
Percolation on finite irregular trees
Consider a rooted tree of height $h$, such that all the leaves are at last layer. We perform the following random process: each edge is deleted with probability $0.5$, and otherwise it is retained. We ...
- 419
3
votes
0
answers
89
views
Two game-set inequalities
Here are a couple of curious related results about a generalized 2-player 1-set tennis game: the winner of the set is the first player to win $n$ games, and the winner of each game is the first player ...
- 1,190
6
votes
0
answers
105
views
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^...
8
votes
1
answer
198
views
Tail bound of a distribution
Let $X_1, X_2, \ldots, X_n, Y_1, Y_2, \ldots, Y_n$ be independent binary random variables each being $1$ with probability $\frac{1}{k}$.
Let $Z = X_1(Y_1 + \cdots + Y_k) + X_2(Y_2 + \cdots + Y_{k+1})...
- 305
6
votes
1
answer
370
views
Mean minimum distance for M and N uniformly random points on reals between 0 and 1
Similar to Mean minimum distance for N random points on a one-dimensional line, but instead of only N random points, choose N and M random points and find the mean minimum distance between points of N ...
