Questions tagged [random-permutations]
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52
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Non-adjacent permutations
Suppose we have an $N$ by $M$ table. Suppose that $x=(a,b)$ and $y=(c,d)$ are two locations in the table, specified by their row and column indexes. We say that (x,y) is horizontally adjacent if $c=...
- 3,879
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1
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209
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Sum of arrival times of Chinese Restaurant Process (CRP)
Suppose that a random sample $X_1, X_2, \ldots$ is drawn from a continuous spectrum of colors, or species, following a Chinese Restaurant Process distribution with parameter $|\alpha|$ (or ...
- 131
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521
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Is the integer factorization into prime numbers normally distributed?
Edit:
Sorry, for the inconvenience: I have edited the question, since there was a misconception in my thinking.
Let $P_1(n) := 1$ if $n=1$ and $\max_{q\mid n, \text{ } q\text{ prime}} q$ otherwise, ...
- 2,608
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99
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Expectation of the operator norm of projection of a random permutation matrix
Assuming I have a fixed dimension $p$ subspace of $\mathbb{R}^d$ orthogonal to $1^d$ and $VV^\top$ with $V \in \mathbb{R}^{d \times p}$ is the orthogonal projection to the subspace.
What bound can I ...
- 23
13
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2
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368
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Expected sorting time of random permutation using random comparators
In sorting networks, a comparator of positions $i < j$ is an operator which takes a permutation, checks if $p_i > p_j$, and if it is the case, swaps $p_i$ and $p_j$.
Using this, we can define ...
- 3,064
1
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1
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What is the function defined by f(k) = #σ1({1,2,…,k})∩σ2({1,2,…,k})∩{1,2,…,k}, where σ1,σ2 are a uniformly random permutations of size N?
Thanks to David Pechersky excellent answer we know that
expectation of $ | σ({1,2,…,k}) ∩ \{1,2,…,k \} | \rightarrow k^2/N$ for σ uniformly random permutation over $N$.
What about the same ...
- 24.4k
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What curve is defined by the formula $f(k) ={}$length of intersection of the first $k$ elements for two random permutations?
Let us fix $N$. Note that function $f$ defined below will satisfy $f(0)=0, f(N) = N$ and it is monotonically increasing (not strictly).
The code for the function seems to me more clear way to ...
- 24.4k
1
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1
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104
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Popular algorithms (stopping rules) with output - a prefix of a permutation
What are some popular settings, when we look at the elements of a randomly generated permutation one by one, and we use certain stopping rule which, as a result gives us a prefix of the observed ...
- 109
4
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90
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(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_\...
- 695
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142
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Random permutations constructed via randomly chosen transpositions
Given positive integers $k$ and $n$, we define the probability distribution $p_{n,k}$ on $S_n$ as:
$$
p_{n,k}(\sigma):=\frac{\#\{(\tau_1,\dots,\tau_s)\mid \sigma=\tau_1\dots\tau_s, \text{ each }\tau_i\...
- 3,554
2
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1
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337
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Law of large numbers for triangular arrays whose moments "look independent"
Let $(X_{n,k})_{k=1,\ldots,n}^{n\in\mathbb{N}}$ be a triangular array of random variables with finite moments of all orders, with no assumptions on their independence. Suppose that
$$
\mathbb{E}\left[\...
- 993
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132
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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 ...
- 1,909
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79
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Relating sequence with or without replacement
I derived a relationship between sequences drawn with and without replacement for an application in genetics. The proof is easy enough, but I would rather find a source than provide a derivation of a ...
- 31
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450
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Random permutations without double rises (avoiding consecutive pattern $\underline{123}$)
A permutation avoiding a consecutive pattern $\underline{123}$ is permutation
$\pi = \pi_1 \pi_2 \ldots \pi_n$ with the property that there does not exists $i \in [1, n-2]$
such that $\pi_i < \pi_{...
- 335
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What probability distribution is this?
Thank you in advance for any suggestions or feedback.
I have a discrete 1D probability distribution represented as a vector $\textbf{p}$, $p_i = p(x_i)$.
I am interested in finding the Wasserstein (...
