All Questions
Tagged with pr.probability co.combinatorics
802 questions
4
votes
1
answer
151
views
Mean occurrences of letters in complete strings given by a Bernoulli scheme
Suppose one has an alphabet of $K$ letters, from which we draw sequentially letters; assume that the $n$-th letter occurs with a fixed probability $p_n$ independently of the others and of the previous ...
- 418
4
votes
1
answer
466
views
Maximum vertical distance for a lattice path when NSEW steps are allowed
Suppose we have a lattice path in 2D starting at the origin, in which north, south, east, and west steps are allowed. For a given path $L$, let $\max(L)$ be the maximum value of the $y$ coordinate ...
- 3,283
4
votes
2
answers
295
views
Distribution of the biggest gap
Randomly select $n$ numbers from the universe $\{1,2\dots,m\}$ without replacement, and sort the numbers in ascending order.
We can get a list of number $\{(a_1,a_2,\dots,a_n\)}$, and then we can ...
- 177
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
301
views
Strings with no long runs from proper subalphabets
Let $R_{n,k,b}$ be the number of $b$-ary strings of length $n$ that contain some run of length at least $k$ from some $(b-1)$-ary subalphabet. Let $N_{n,k,b}=b^n-R_{n,k,b}$ be the size of the ...
- 24.8k
4
votes
1
answer
1k
views
Estimating the distribution of minimal hamming distances within a set of strings?
Is their an efficient mathematical way to estimate the distribution of minimal hamming distances for a set of random strings of length 8 over a 4-letter alphabet? E.g. given a set of 100-10,000 ...
- 85
4
votes
2
answers
380
views
Joint probability distribution as functions
Suppose $X$ and $Y$ are correlated random variables in a finite set ${\mathcal A}$, and let $f, g$ be functions that map elements from ${\mathcal A}$ to ${\mathcal B}$ for some finite set ${\mathcal B}...
- 305
4
votes
3
answers
286
views
Medium-Sized Calculations and Organization
This is not a math question as much as a process question. For the first time in my (very short) career, I find myself doing one of those messy calculations, where each 'line' of the calculation can ...
- 63
4
votes
2
answers
145
views
Understanding equiprobable trinomial identity
With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
- 284
4
votes
1
answer
261
views
What is the convergence rate of this "infinite monkey"-type probability?
Cross-posted from Math Stack Exchange, where it hasn’t received an answer yet:
Let $S$ be a finite set and $n,m\in\mathbb N$. Consider the process $R=(R_i)_{i\in\mathbb N}$ where all $R_i$ are iid ...
- 141
4
votes
3
answers
324
views
Probability that $k$ random subsets of a fixed size covers a set
Let $A=\{1,\ldots,n\}$. Now, we uniformly randomly select $k$ subsets, $A_i$ of size $d$ from $A$. What is the probability that $\bigcup_i A_i=A$? This seems to be natural variant of the set cover ...
- 83
4
votes
1
answer
264
views
Information for discovering an item-colour assignment in a combinatorial game
We are given a set $S=\{i_1, i_2, \ldots, i_n\}$ of items and a set $C=\{c_1, c_2, \ldots, c_m\}$ of colours. Each item in $S$ is tinted with one colour $c\in C$. Let $\mathcal{A}$ be the set of all ...
- 1,677
4
votes
1
answer
299
views
Collecting stones in n buckets
There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new ...
- 1,269
4
votes
2
answers
314
views
Convexity of truncated expectation
Let $k, n$ be two positive integers with $k \leq n$, and let $P = \{ (x_1, \dots, x_n) \in [0, 1]^n : \sum_i x_i = k \}$.
Given $x = (x_1, x_2, \dots, x_n) \in P$, let $X_i$ be the random variable ...
- 41
4
votes
1
answer
421
views
Order statistic of Markov chain sample path and related probabilities
Consider a one dimensional sample path, denoted as $\{X(1), ..., X(t), ..., X(n)\}$, generated from a discrete time finite state (time homogeneous) Markov chain over states $\{1,...,m\}$, with ...
- 1,538
4
votes
1
answer
275
views
Nontrivial lower bounds on Cheeger inequalities for Markov chains
For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the size of $K$ when ...
- 51
4
votes
2
answers
420
views
Generating a group by randomly sampling generators
Let $G$ be a finite abelian group, $n$ a positive integer and let $G^n$ denote the direct product of $n$ copies of $G$. We say an element of $G^n$ is full if it acts as a nonidentity element of $G$ in ...
- 2,649
4
votes
1
answer
232
views
Negative Association of Component Size in Random Hypergraph
I have a $d$-uniform hypergraph on $n$ vertices with $k$ hyperedges, where $d << k$ and $n = 4k d^2$ or so.
The hyperedges are placed independently uniformly at random. I would like to have a ...
- 241
4
votes
0
answers
145
views
Is it easier to exit a box to the right of a box in $\mathbb{Z}^2$ if I remove some edges to the left?
Suppose that I am given the graph $G = (V,E)$ where
$V = \{ 1, 2, \dots 2N+1 \} \times \{ 1, 2, \dots 2N+1 \} $
and there is an edge between two vertices $(n,m)$ and $(n',m')$ if and only if
$\vert n-...
- 1,054
4
votes
0
answers
1k
views
Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
- 47
4
votes
0
answers
414
views
Explicit formula for tournament sequence
I am looking for an explicit formula for a sequence. The sequence is generated as follows:
There is a tournament with $10$ teams. In the beginning, all teams have a 0-0 win-loss record. The teams are ...
- 41
4
votes
0
answers
93
views
(Asymptotic) Cycle structure in a random permutation given total number of cycles?
A random $N$-permutation is the one drawn uniformly from all possible permutations on $N$ points.
We know that the expected number of cycles of length $\ell$ in a random $N$-permutation, $\mathbb{E}C_\...
- 715
4
votes
0
answers
144
views
Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
- 1,677
4
votes
0
answers
182
views
Determine the minimal elements of a Dynkin system generated by a finite set of finite sets
(This is a refined version of https://cs.stackexchange.com/q/144371)
Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power set of $\Omega$ containing $\Omega$, which is ...
- 5,822
4
votes
0
answers
187
views
Dyadic distribution of $0/1$ permanents
Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$.
What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents
in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...
- 13.9k
4
votes
0
answers
216
views
How frequent are permutations with small interleaving?
For $S\subset [n]:=\{1,2,\dotsc,n\}$, define $\delta(S)$ to be the number of $m\in S$ such that $m+1\notin S$.
Let $\pi$ be a permutation of $[n]$. For simplicity, assume that $\pi$ is an $n$-cycle. ...
- 20.2k
4
votes
0
answers
541
views
How should the proof of the XYZ theorem be understood?
The XYZ Theorem of Shepp [1] states the following for a given poset $P$. Consider the probability space of all the linear extensions of $P$, where each possible extension is equally likely. Then for ...
- 571
4
votes
0
answers
266
views
Metrics on finite groups and generalizations of central limit theorems for balls volumes (à la Diaconis-Graham)
In wonderful lectures by P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" metrics on permutation groups are considered and ...
- 24.9k
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
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
4
votes
0
answers
150
views
Dividing a finite arithmetic progression into two sets of same sum: always the same asymptotics?
This is inspired by the recent question How many solutions $\pm1\pm2\pm3…\pm n=0$.
The oeis entries A063865 linked to this question and A292476/A156700 for the related one "How many solutions $\pm1\...
- 13.4k
4
votes
0
answers
188
views
Distributions over permutation groups $\mathcal{S}_n$
Partly inspired by recent developments in enumeration of pattern avoiding permutations, which is known to be connected with Brownian excursions [Hoffman&Rizzolo]. The exciting milestone is the ...
- 8,071
4
votes
1
answer
398
views
Variance of load in maximally loaded bin, if $m$ balls are thrown into $n$ bins
In the paper "Balls and bins a simple and tight analysis" by Raab and Steger, available here strong upper and lower bounds are proved about the number $M$ of balls in a maximally loaded bin when $m$ ...
- 10.4k
4
votes
0
answers
83
views
The max of a random sum, SK model
Let $(N_{i,j}, i,j \in \mathbb N)$ be independent (standard) Gaussian random variables.
What is known/ what is conjectured /what can we say about
$$\max \lbrace{\sum_{1 \leq i,j \leq n} \epsilon_i \;...
- 468
4
votes
0
answers
141
views
Level sets of function of inner products of vectors on hypercube
Let $H = \{ 0, 1\}^d$ be the $d$-th Cartesian product of $\{0, 1\}$ in $\mathbb{R}^d$. Suppose $v_1, \ldots, v_k$ are $k$ vectors in $H$ in general position. We define function $F \colon H^{k}\...
- 1,127
4
votes
0
answers
107
views
Random polyominoes containing $2\times2$ squares
The construction quoted in the question "How to sample a uniform random polyomino?" claims to produce a "uniform random polyomino". But apart from the mentioned possibility of getting stuck, it also ...
- 13.4k
4
votes
0
answers
94
views
Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius
I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...
- 161
4
votes
0
answers
220
views
Navigation in a graph
The problem
Let $G=(V,E)$ be a graph. $k = O\left(\log(|V|)\right)$ distinct vertices are picked randomly from $V$. We call the set of chosen $k$ vertices $T$.
Assumptions about the graph: You may ...
- 323
4
votes
0
answers
492
views
Convex hull of a discrete set of points
If i was to give an $n×n$ grid with each grid point having probability $p$ of being selected, would it be difficult to calculate distributions of various measures regarding the convex hull of all ...
- 903
4
votes
0
answers
133
views
Generating random weak k-bounded reverse plane partitions
Fix a partition $\lambda$. A weak reverse plane partition of shape $\lambda$ is a filling $0\leq \pi_{ij}$ of $\lambda$ with $\pi_{ij}\leq \pi_{kl}$ whenever $i\leq k$ or $j\leq l$. Note that $\pi_{ij}...
- 4,952
4
votes
0
answers
1k
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Matula-Goebel ordering of rooted trees intrinsic?
I was somewhat recently introduced to the Matula-Goebel bijection between rooted trees and natural numbers. (nicely illustrated here http://keithbriggs.info/matula.html) Looking through them, I ...
- 475
4
votes
0
answers
114
views
Bounds on the size of a set of strings over an arbitrary alphabet within a fixed Hamming distance of one-another
I pick a set of random strings $S$ of length $L$ over an $P$-letter alphabet. These strings are 'random' in the sense that every character is chosen with uniform random probability over the ...
- 41
4
votes
0
answers
216
views
How should one generate a random set of mappings?
My motivation for this question comes from the study of synchronizing automata. There is a general consensus that random automata are synchronizing and have short synchronizing words. I am hoping ...
- 38.6k
4
votes
0
answers
117
views
symmetric difference of temperate zone and inscribed disk
For random domino tilings of the Aztec diamond of order $n$ or random lozenge tilings of the regular hexagon of order $n$, what's the typical order of magnitude of the area of the symmetric difference ...
- 19.7k
4
votes
0
answers
331
views
What is 'arch' in Vershik-Kerov's 1984 paper?
In their 1984 paper Asymptotic of the Largest and the Typical Dimensions of Irreducible Representations of a Symmetric Group, Vershik and Kerov use the notation $\DeclareMathOperator{\arch}{arch}\arch ...
- 445
4
votes
0
answers
580
views
Monotonic properties of harmonic functions on graphs
I have a question concerning monotonic properties of "generalized harmonic functions" on graphs. I am a physicist and I didn't take any separate courses in neither graph theory nor discrete harmonic ...
- 965
4
votes
0
answers
497
views
A Local CLT with large variance
For n an even integer, $0 \leq i \leq$ ${n} \choose{j}$, $1 \leq j \leq n$ let $X_{i,j}$ be a
random variable taking values $\frac{n}{2}-j,0,j - \frac{n}{2}$ with equal probability. Let $S_{n}$ be ...
- 263
4
votes
1
answer
839
views
A balls into bins problem with combinatorial constraints
We are given $m$ balls and $n$ bins, with $m \ge n$. Each bin can contain at most $c$ balls (we assume that $c$ is an even integer). In a sequential fashion, at each time step, one ball is placed into ...
- 1,677
3
votes
7
answers
4k
views
How to tell if two random polynomials are identical
Let t be a positive real number. Let P(x) and Q(x) be two random polynomials with integer coefficients. If P(t) = Q(t), then what is the probability that P(x) is not identical to Q(x)?
Will it make a ...
- 179
3
votes
1
answer
406
views
Moments of a random variable related to uniform distribution on sphere
Let $u$ be taken uniformly from the unit sphere $\mathbb S^{n-1}$ and $D$ be a diagonal matrix. I'd like to find a general formula for
$$
\mathbb E[(u^\top D u)^m]
$$
for $m=1,2,3, \dots$, in terms of ...
- 1,608