(42S) Subfactorial

[C.Ret]

(09-06-2021 06:50 PM)John Keith Wrote:  [...]The second formula can also be simplified to round(n!/e). More information and formulas at A000166.

On Free42 or the DM42 you have over 30 digits of precision so your first program (either formula) will be exact for fairly large values of n.

The round trick give me the idea for a short and fast code for my HP-15C :

   

Quote:In[2]:= subfac[30]
Out[2]= 97581073836835777732377428235481

30 f D compute !30 and f PREFIX display 97.58107385 E 30 these makes 9/10 digits are alright, fast and not bad for a truthfully Voyager oldie !

5 f D display exactly 44, let verify the statistics :
12345 - 21345 - 31245 - 41235 - 51234 *
12354 - 21354 - 31254 * 41253 * 51243 -
12435 - 21435 - 31425 - 41325 - 51324 -
12453 - 21453 * 31452 * 41352 - 51342 -
12534 - 21534 * 31524 * 41523 * 51423 *
12543 - 21543 - 31542 - 41532 * 51432 *
13245 - 23145 - 32145 - 42135 - 52134 -
13254 - 23154 * 32154 - 42153 - 52143 -
13425 - 23415 - 32415 - 42315 - 52314 -
13452 - 23451 * 32451 - 42351 - 52341 -
13524 - 23514 * 32514 - 42513 - 52413 -
13542 - 23541 - 32541 - 42531 - 52431 -
14235 - 24135 - 34125 - 43125 - 53124 *
14253 - 24153 * 34152 * 43152 * 53142 -
14325 - 24315 - 34215 - 43215 - 53214 *
14352 - 24351 - 34251 * 43251 * 53241 -
14523 - 24513 * 34512 * 43512 * 53412 *
14532 - 24531 * 34521 * 43521 * 53421 *
15234 - 25134 * 35124 * 45123 * 54123 *
15243 - 25143 - 35142 - 45132 * 54132 *
15324 - 25314 - 35214 * 45213 * 54213 *
15342 - 25341 - 35241 - 45231 * 54231 *
15423 - 25413 * 35412 * 45312 - 54312 -
15432 - 25431 * 35421 * 45321 - 54321 -

Total :
5! = 120 arrangements (- & *)
!5 = 44 derangements ( * )
How the asterisk are randomly distributed makes me believe of un vrai dérangement.

Couldn't Albert Chan please verify that with 30 elements, he exactly gets 97581073836835777732377428235481 asterisks (*) ?