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I'm falling in love with random permutations.
The average length of the longest cycle in a random permutation of a huge n-element set approaches the "Golomb-Dickman constant" times n.
(1/n)
The average length of the longest cycle in a random permutation of a huge n-element set approaches the "Golomb-Dickman constant" times n.
(1/n)
The Golomb-Dickman constant also shows up in number theory... in a very similar way!
If you randomly choose a huge n digit integer, the average number of digits of its largest prime factor is asymptotically equal to the Golomb-Dickman constant times n.
(2/n)
If you randomly choose a huge n digit integer, the average number of digits of its largest prime factor is asymptotically equal to the Golomb-Dickman constant times n.
(2/n)
So, there's a connection between prime factorizations and random permutations!
You can read more about this in Jeffrey Lagarias' paper about Euler's constant:
https://arxiv.org/abs/1303.1856
The Golomb-Dickson constant seems to be a relative of Euler's constant.
(3/n)
You can read more about this in Jeffrey Lagarias' paper about Euler's constant:
https://arxiv.org/abs/1303.1856
The Golomb-Dickson constant seems to be a relative of Euler's constant.
(3/n)
One more thing!
Say you randomly choose a function from a huge n-element set to itself. The average length of its longest periodic orbit asymptotically equals the Golomb-Dickman constant times the square root of π/2 times the square root of n.
(4/n)
Say you randomly choose a function from a huge n-element set to itself. The average length of its longest periodic orbit asymptotically equals the Golomb-Dickman constant times the square root of π/2 times the square root of n.
(4/n)
