Monday, September 6, 2021

Swiss Micros DM42: Subfactorial and Numworks Update (16.3)

Swiss Micros DM42: Subfactorial


Happy Labor Day!


This is a request by Marko Draisma and gratitude to Mr. Draisma.


Calculating the Subfactorial


A common,  and perhaps the most straight forward, formula to calculate the subfactorial is:  


!n = n! × Σ((-1)^k ÷ k!, k=0 to n)


Yes, the subfactorial is written with the exclamation point first.  The subfactorial finds all the possible arrangements of a set of objects where none of the objects end up in their original position.


For example, when arranging the set {1, 2, 3, 4} the subfactorial counts sets such as {2, 1, 4, 3} and {3, 4, 1, 2} but not {1, 4, 3, 2}.  For the positive integers:   !n < n!.


I am going to present two programs.  The first will use the formula stated above.


The second uses this formula, which will not require recursion or loops:


!n = floor[ (e + 1/e) × n! ] - floor[ e × n! ]


Note: Since the N! function on the DM42 accepts only positive integers, we can use the IP (integer part) to simulate the floor function.


integer(x) = { floor(x) if x ≥ 0,  ceiling(x) if x < 0


The following programs can be used on Free42, HP 42S, or Swiss Micros DM42.


Swiss Micros DM42 Program:  Subfactorial Version 1


This is a traditional route.  Registers used:


R01:  k,  counter

R02:  sum register 

R03:  n!, later !n


Program labels can start with symbols on the 42S.


01  LBL "!N"

02  STO 01

03  N!

04  STO 03

05  0

06  STO 02

07  RCL 01

08  1E3

09  ÷

10  STO 01

11  LBL 00

12  RCL 01

13  IP

14  ENTER

15  ENTER

16  -1

17  X<>Y 

18  Y↑X

19  X<>Y

20  N!

21  ÷

22  STO+ 02

23  ISG 01

24  GTO 00

25  RCL 02

26  RCL× 03

27  STO 03

28  RTN


Swiss Micros DM42 Program:  Subfactorial Version 2


I only put 2 in the label to distinguish the two programs.  


01  LBL "!N 2"

02  N!

03  ENTER

04  ENTER

05  1

06  E↑X

07  ENTER

08  1/X

09  +

10  ×

11  IP

12  X<>Y

13  1

14  E↑X

15  ×

16  IP

17  -

18  RTN



Examples


!2 = 1

!3 = 2

!4 = 9

!5 = 44

!9 = 133,496

!14 ≈ 3.2071E10


Sources


"Calculus How To:  Subfactorial"   College Help Central, LLC .https://www.calculushowto.com/subfactorial/ Retrieved September 5, 2021. 



Weisstein, Eric W. "Subfactorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Subfactorial.html  Retrieved September 5, 2021


Numworks 16.3 Update

Numworks recently updated its firmware to Version 16.3.  Find details of the changes and additions here:

https://my.numworks.com/firmwares

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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