[r-t] Extents of minor
Philip Saddleton
pabs at cantab.net
Wed Aug 18 21:44:38 UTC 2004
Stephen Penney <stephen at ucalegon.com> wrote at 10:35:55 on Wed, 18 Aug
2004
>Simple question: how many extents of minor are there?
>
A probabilistic approximation: if there are n rows and k place
notations, there are n! permutations of the rows. The probability that a
random pair of adjacent rows differ by a valid change is (k/(n-1)). If
these probabilities were independent (which they are not), the number of
extents is approx
n! x (k/(n-1)) ^ n
giving
Singles n=6, k=2: 2.9
Minimus n=24, k=4: 3.6 x 10^5
Doubles n=120, k=7: 10^51
Minor n=720, k=12: 10^466
This appears to be an overestimate for small numbers
Alternatively, there are k^n sequences of place notation. The
probability that the ith row differs from the preceding i (including the
initial rounds) is (n-i)/n. The nth row must be rounds, so the number of
extents is approx
(k/n)^n x (n-1)!
giving
Singles: 0.16
Minimus: 5.4 x 10^3
Doubles: 10^49
Minor: 10^463
This appears to be an underestimate for small numbers
Maybe the answer is somewhere in between.
--
Regards
Philip
http://www.saddleton.freeuk.com/
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