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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)pic.twitter.com/petzzADSID
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)
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)
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)
How would one even attempt to prove the rationality/irrationality of the Golomb-Dickman constant?
It's gonna be irrational. But if one wants to prove that, start by learning how proved first e, then pi, then zeta(3) were irrational. These are in order of increasing difficulty. Then, as a warmup exercise, prove that Euler's constant is irrational (famous unsolved problem).
They are closely connected, since a permutation determines a partition (given by its cycles). So, we can think of random permutations as giving another probability distribution on the set of partitions.
Nice. I need to study up more on Golomb's work. I developed a love-hate relationship with Golomb-Rice codes after implementing a JPEG2000 image codec and an unrelated lossless video codec.
What's the hate part? Golomb also invented polyominos: https://en.wikipedia.org/wiki/Polyomino maximum length shift register sequences: https://en.wikipedia.org/wiki/Maximum_length_sequence … and many other things!
I know Golomb, I've spent some time studying his rulers, but who is this Dickman? Wikipedia just failed me :-( :-o 8-)
It seems like Dickman was a number theorist, since there's a formula for the Golomb-Dickman constant in terms of the Dickman function: https://en.wikipedia.org/wiki/Dickman_function … and both are connected to the average behavior of prime factorizations.
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