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PrimeProductsMinusOne on Permutation ​

Positions i with i * p(i) - 1 prime.

On Permutation

Called as CombinatorialStat(x, "PrimeProductsMinusOne"). The head's page has its examples.

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FindStat St001287

The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.

ObjectValue
[1]0
[1,2]1
[2,1]0
[1,2,3]1
[1,3,2]2
[2,1,3]0
[2,3,1]2
[3,1,2]2

Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001287 has the rest.