All Questions
Tagged with co.combinatorics pr.probability
802 questions
6
votes
3
answers
855
views
Series involving power of the index
How to prove the following identity
$$ \sum_{n=1}^{\infty} \frac{n^{n-1} e^{-n}}{n!} = 1$$
analytically (which can be confirmed with $Mathematica$)? The standard trick for geometrical series does not ...
- 77
1
vote
0
answers
123
views
On probability of coprimality of a list of numbers
We know $r$ randomly chosen integers are coprime with probability $\frac1{\zeta(r)}$.
Pick a bound $N$ and pick $k\lceil N^{\alpha}\rceil$ uniformly random integers in $[0,N^{\alpha+\beta}]$ where $\...
- 13.9k
7
votes
2
answers
366
views
On permanent of a square of a doubly stochastic matrix
Let $A = (a_{i,j})$ be a double stochastic matrix with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A permanent of a matrix $A = (a_{i,j})$ is ...
user125543
1
vote
1
answer
183
views
Expectation of edge weights on the complete graph
Let $n,k \geq 3$ be positive integers with $n$ much larger than $k$ and consider a random assignment of weights to the edges of the complete graph $K_n$. On each vertex of $K_n$ we attach a random ...
- 26.9k
3
votes
1
answer
135
views
Cycle counts in Ewens measure as $\theta$ diverges
For $w$ a permutation, let $c(w)$ denote the total number of cycles and $c_i(w)$ denote the number of $i$-cycles.
The Ewens measure is a one-parameter probability distribution on permutations where ...
- 1,989
6
votes
0
answers
99
views
q-binomial-like series with exponentials defining probability distribution
Recently I encountered the series
$$f(d) = \frac{1}{(t;t)_\infty} \sum_{k=0}^\infty \frac{(-1)^k t^{\binom{k}{2}}}{(t;t)_k} e^{-t^{d-k}}$$
where $(t;t)_n = \prod_{i=1}^n (1-t^i)$, and $0 < t < 1$...
- 568
4
votes
3
answers
269
views
Existence of (near) equidistant codewords
My question is originally related to coding theory, but fairly easy to state in pure combinatorial way.
Fix $k\in\mathbb{N}$, $\beta\in(0,1)$ and consider the binary cube $\Sigma_n = \{0,1\}^n$ ...
- 1,096
2
votes
1
answer
383
views
Lower bound and limit of a sum with binomial coefficients
Let
$$A_k = \sum_{i=1}^k i {3k-2i-1 \choose i-1} {2i-2 \choose k-i}$$
$$B_k = \sum_{i=1}^k i {3k-2i-2 \choose i-1} {2i-1 \choose k-i}$$
$$C_k = \sum_{i=1}^k (3k-2i-2) {3k-2i-3 \choose i-1} {2i\...
- 155
5
votes
4
answers
917
views
Limit of a sum with binomial coefficients
Let $$A_k = \frac{\sum_{i=1}^ki{2k-i-1 \choose i-1}{i-1 \choose k-i}}{k{2k-1\choose k}}$$
$$B_k = \frac{\sum_{i=1}^ki{2k-i-2 \choose i-1}{i \choose k-i}}{k{2k-1\choose k}}$$
$$C_k = \frac{\sum_{i=1}^k(...
- 155
0
votes
0
answers
58
views
Impact of reducing the number of distinct elements in the Count distinct problem
I am dealing with the Count distinct problem and Space saving algorithm. The problem goes like that:
I have a stream of $N$ elements. The number of distinct elements is $D$. Space saving algorithm is ...
- 11
1
vote
2
answers
302
views
Counting permutations defined by a simple process
Consider $n$ labeled balls, $k$ of which are red and $(n-k)$ blue. Given a permutation of these balls, we tick $n-1$ times. For the $i$-th tick, if the $i$-th ball in the permutation is red, then it ...
- 155
8
votes
0
answers
304
views
"Meritocratic" pyramid schemes
There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved ...
- 2,075
1
vote
0
answers
122
views
Probability puzzle on partitions
Consider a set $U$ of size $n$ and let $\mathcal{S}$ be the set of all $(n/2)$-subsets of $U$ (assume $n$ is divisible by 4). Let $P$ be a partition of $\mathcal{S}$ into $k$ blocks $B_1,\dots,B_k$.
...
- 11
17
votes
1
answer
1k
views
Can this probability be obtained by a combinatorial/symmetry argument?
Suppose that $a_1,\dots,a_n,b_1,\dots,b_n$ are iid random variables each with a symmetric non-atomic distribution.
Let $p$ denote the probability that there is some real $t$ such that $t a_i \ge b_i$ ...
- 128k
2
votes
0
answers
138
views
Update on Viskov's paper on random processes, Lagrange inversion, and the Heisenberg–Weyl algebra
"A Random Walk with a Skip-Free Component and the Lagrange Inversion Formula" by Viskov presents connections among Lagrange inversion and measures of random Lévy processes. The freely ...
- 10.5k
2
votes
1
answer
165
views
Is a random $(r+1,r)$-biregular bipartite graph $r$-edge connected w.h.p?
A uniformly random $r$-regular bipartite graph on $n$ vertices is known to be $r$-edge connected. That is, with high probability as $n$ grows large, the minimum size of a cut in a random $r$-regular ...
- 153
4
votes
1
answer
225
views
What is the generalization of the formula for Chung and Feller's Theorem 2 to odd numbers of steps?
In their classical paper on fluctuations in coin tossing On Fluctuations in Coin-Tossing, Chung and Feller give a precise formula for the conditional probability of the number of positive “sides” of a ...
- 471
9
votes
1
answer
889
views
Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory
In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
- 10.5k
1
vote
0
answers
663
views
The distribution of hitting time in 2D-lattice random walk [closed]
Assume a particle at $(0,0)$ with the same possibility of $1/4$ for moving up/down/left/right (i.e. random walk in 2D lattice). We define the stopping time 𝑇𝑐 as it hits $(a,b)$. How can we get the ...
3
votes
0
answers
516
views
The distribution of collision stopping time in 2D random walk
Assume two particles A at $(0, 0)$ and B at $(a, b)$ in 2D discrete grid, both of them have the same possibility of $\frac{1}{4}$ for moving up/down/left/right (i.e. 2D random walk). We define the ...
4
votes
1
answer
322
views
Approximating binomial coefficient sum
I have the following exact sum for the expectation of an event
$$\sum_{m=0}^{nk} \sum_{j=1}^n (-1)^{j-1}\binom{n}{j} \binom{(n-j)k}{m} / \binom{nk}{m}$$
which is exactly correct but I want to give an ...
- 51
12
votes
3
answers
909
views
Expected number of compositions needed to get constant function
This is somewhat inspired by Factoring a function from a finite set to itself.
Fix natural number $n$ and let $[n] := \{1,2,\ldots,n\}$. Set $g_0 \colon [n]\to [n]$ to be the identity, and for $i \geq ...
- 24.2k
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
3
votes
1
answer
271
views
A quantity associated to a probability measure space
Let $(S,P)$ be a (finite) probability space. We associate to $(S,P)$ a quantity $n(S,P)$ as follows:
The probability of two randomly chosen events $A,B\subset S$ being independent is denoted by $n(S,P)...
- 356
4
votes
1
answer
262
views
What is the number of finite Dynkin systems?
(This is a spin-off of Determine the minimal elements of a Dynkin system generated by a finite set of finite sets)
Let $\Omega$ be a finite set. A Dynkin system on $\Omega$ is a subset of the power ...
- 5,822
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
2
votes
1
answer
158
views
"Shape"/"norm" of a uniformly random set partition
Let $\mathcal{A}=\{A_1, A_2, \ldots, A_m\}$ be a uniformly random set partition of $[n]$.
What can we say about $||\mathcal{A}||_2 = \sqrt{\sum_{i=1}^m |A_i|^2}$? It is clearly upper bounded by $n$, ...
- 526
3
votes
1
answer
162
views
Recurrence relation for the moments of the GOE
The Harer-Zagier formula provides a three term recurrence relation for the expected value of the single-trace operator $\mathrm {Tr}(X^k)$ where $X$ is a $N\times N$ matrix from the GUE. Is there an ...
1
vote
1
answer
124
views
Empirical degree distribution of random $n$ vertices labeled rooted tree converges to Poisson distribution
I am reading Louigi's lecture note on random trees and graphs here. I get stuck on part (b), Exercise 1.2.3 on page 19, which says the following:
Let $T_n$ be uniformly drawn from $\mathcal{T}_n$, ...
- 715
5
votes
2
answers
707
views
Distribution of some sums modulo p
Fix a finite set of integers $S$ and a prime number $p$. Let $(a_1, a_2, \dotsc, a_n)$, $(b_1, b_2, b_3, \dotsc, b_n)$ be two sequences of integers where the numbers $a_i$ and $b_i$ are chosen ...
- 1,101
8
votes
1
answer
380
views
Question about estimating random symmetric sums modulo p
Let $n > 0$ be a positive integer (large) and $p > 2$ a fixed prime number. What is the probability that $$\sum_{ 1 \leq i < j \leq n} a_ia_j = 0 \mod p$$ where $a_1, a_2, \dots a_n$ are ...
- 1,101
3
votes
1
answer
118
views
Using singularity analysis for probability at a threshold?
In some urn model with parameter $p$, the generating function
$$
f_p(z) \;=\; \frac{1+p\,z}{1-(1-p)\,z\,(1+p\,z)}
$$
is such that $[z^n]f_p(z)$ is the probability that an $n$-urn configuration has a ...
- 674
3
votes
1
answer
241
views
Probability that k randomly drawn permutations can be arranged to compose to the identity
Consider the symmetric group $S_n$ under the uniform distribution. For integer $k > 1$, suppose we draw $k$ elements $s_1, \dots, s_k$ independently at random. What is the probability that there ...
- 6,155
11
votes
2
answers
1k
views
Heuristic lower bounds on small sums of roots of unity
Let $f(k,n)$ be the smallest non-zero absolute value of a sum of $k$ complex $n$th roots of unity. Asking for bounds in either direction, Tao suggested that a polynomial lower bound seemed plausible ...
- 4,589
3
votes
1
answer
161
views
Probability permutation in turned to cycle
Let $M$ be a $0/1$ square matrix having one $1$ per row and column (permutation matrix).
If you permute the columns and rows independently what is the probability resulting permutation matrix is a ...
- 13.9k
1
vote
0
answers
47
views
Probability that a modified $c$-bounded lattice walk is also $c$-bounded
Let $\mathcal{B}_n^{(c)}$ be the set of $n$ step lattice bridges (starts at $(0,0)$ ends at $(n,0)$), taking steps $\{(1,1), (-1,1)\}$ bounded between $y = c$ and $y = -c$ for a constant $c \geq 0$. ...
- 111
0
votes
1
answer
217
views
On independence of multiples of $\mathbb Z_p$
This is a rewording in combinatorial language of a question posed on another forum. The original was posed as a probabilistic problem.
Problem set up:
Consider for a fixed prime $p$, the ...
- 6,155
13
votes
2
answers
518
views
Asymptotics of a randomized Fibonacci sequence
Let $f(1)=f(2)=1$ and recursively define $f(n+1) = f(n) + f(i)$, where $i$ is chosen uniformly at random from $1,\ldots,n-1$. About how big should we expect $f(n)$ to be for $n$ large? We can examine ...
1
vote
1
answer
119
views
Does exponential degree distribution entail Log-normal distance distribution in large complex graphs?
We've been exploring the graph structure of a large genealogical data base (WikiTree) of which main connected component contains about 23 million nodes. The graph edges are defined by any direct ...
1
vote
0
answers
176
views
Gaussian order statistics
Setup. Let $\alpha\in(0,1)$ fixed; and $\tau\in[0,1]$ (think of it very close to one).
Suppose $X_1,\dots,X_n$ are i.i.d. standard normal.
Let $Y_1,\dots,Y_n$ be another sequence of standard normals ...
- 11
16
votes
3
answers
2k
views
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 ...
- 163
5
votes
1
answer
325
views
Bounding the max-loaded bin using${m \choose k} \|A\|_k^k$
Throw $m$ balls into $n$ bins independently, each ball selecting a bin from the distribution $A \in \Delta_n$. This question is about lower-bounding the max-loaded bin.
Background. In this MO answer I ...
- 4,529
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
1
vote
1
answer
173
views
Could you provide some TSP examples from real world to test a new algorithm?
It's well known that to find a hamilton cycle is NPC, while TSP is NPH.
But it seems that for majority of graphs (density of edge > 0.1, order > 100) there is a fast algorithm to find different ...
- 175
1
vote
1
answer
207
views
Expectation of the sum of the squares of the cardinal of an inverse function
I sample a random one-to-one function $\pi:\{0\,;\,1\}^n\to\{0\,;\,1\}^n$. I define $f$ as:
$$\forall x\in\left[0\,;\,2^n-1\right]\cap\mathbb{N},f(x)=x\oplus\pi(x)$$
where $\oplus$ is the bitwise XOR.
...
- 167
3
votes
1
answer
128
views
A ratio of two probabilities
I am concerned about the monotonicity of the following ratio
$
f(\eta)=\frac{\sum_{x=K}^{N}\left(\begin{array}{c}
N\\
x
\end{array}\right)\left(q_{G}\eta\right)^{x}\left(1-q_{G}\eta\right)^{N-x}}{\...
- 207
1
vote
1
answer
141
views
Expectation of maximum of all period lengths of functions $f:\{1,\ldots,n\}\to\{1,\ldots,n\}$
This is based on an older question.
For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ and ''starting value'' $...
- 48.9k
6
votes
2
answers
274
views
Expectation of period length of functions $f:\{1,\ldots,n\}\to \{1,\ldots,n\}$
For $n\in\mathbb{N}$, let $[n]:= \{1,\ldots,n\}$. Let $\text{Fun}(n)$ denote the set of all functions $f:[n]\to[n]$. To $f\in\text{Fun}(n)$ associate a sequence $\text{seq}(f))$ defined recursively by
...
- 48.9k
1
vote
1
answer
106
views
Almost-parallel corners of the hypercube in high dimensions
Say I would like a collection of k "almost-parallel" boolean vectors $X_1,...,X_k \in \{\pm 1\}^n$, such that $(X_i,X_j)/n \approx 1-\epsilon$ for some small $\epsilon$. How many ways are ...
- 435
5
votes
3
answers
601
views
Convergence speed of a random dyadic rational generator
We are given a multiset $M$ of real numbers which initially is equal to $\{0,1\}$. In a sequential fashion, at each round $r\in\mathbb{N}$
two distinct instances $x_r$ and $y_r$ of $M$'s numbers are ...
- 1,677