The union of two random permutations does not have a directed Hamilton cycle

This study examined the anecdotal notion that people choose partners based on preferred characteristics that constitute their ‘type’. We gathered the eye colours of participants’ partners across their entire romantic history in three samples (student-centred, adult, and celebrity). We calculated the proportion of partners’ eye colours, and compared that to 100,000 random permutations of our observed dataset using t-tests. This was to investigate if the eye colour choices in the original datasets had greater consistency than in the permutations. Across all samples, we observed no good evidence that individuals make consistent eye colour choices, suggesting that eye colour may not be a priority when choosing a partner.

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Permutations with equal orders

Combinatorics Probability Computing ◽

10.1017/s0963548321000043 ◽

2021 ◽

pp. 1-11

Author(s):

Huseyin Acan ◽

Charles Burnette ◽

Sean Eberhard ◽

Eric Schmutz ◽

James Thomas

Keyword(s):

Random Permutations

Abstract Let ${\mathbb{P}}(ord\pi = ord\pi ')$ be the probability that two independent, uniformly random permutations of [n] have the same order. Answering a question of Thibault Godin, we prove that ${\mathbb{P}}(ord\pi = ord\pi ') = {n^{ - 2 + o(1)}}$ and that ${\mathbb{P}}(ord\pi = ord\pi ') \ge {1 \over 2}{n^{ - 2}}lg*n$ for infinitely many n. (Here lg*n is the height of the tallest tower of twos that is less than or equal to n.)

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Counting Hamilton cycles in Dirac hypergraphs

Combinatorics Probability Computing ◽

10.1017/s0963548320000619 ◽

2020 ◽

pp. 1-23

Author(s):

Stefan Glock ◽

Stephen Gould ◽

Felix Joos ◽

Daniela Kühn ◽

Deryk Osthus

Keyword(s):

Hamilton Cycle ◽

Uniform Hypergraph ◽

Similar Estimate ◽

Hamilton Cycles

Abstract A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

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