algorithmuscanorj

On the invariance of the terms in A215940 (final)

Jan 7th, 2013
676
Never
Not a member of Pastebin yet? Sign Up, it unlocks many cool features!
  1. (With corrections added at the end.)
  2.  
  3.     Greetings,
  4.      
  5.     Dear reader(s), your attention please...
  6.      
  7.     In the moment of writing this to all of you, I have on my table
  8.     a scientific calculator model fx-570ES from CASIO. It allows
  9.     to write an Algebraic expression and evaluate it in 4 different
  10.     bases.
  11.      
  12.     I will give a practical demonstration about what is the actual
  13.     meaning of talking about the terms of A215940 as an "Universal"
  14.     sequence. Let me emphasize now that this is not about the
  15.     quantity of terms or the size of the set of permutations.
  16.  
  17.     First than all, the actually important fact is the divisibility
  18.     of the difference between two permutations by (10-1) for every
  19.     radix or base of a positional number system where such
  20.     permutations could be read.
  21.    
  22.     About the universality of the terms generated by
  23.     quotients of the kind (B-A)\(10-1), for two permutations
  24.     (A<B):
  25.  
  26.     Actually it means an invariant property of those terms
  27.     consisting in being written always with the same numerals
  28.     regardless the radix where they are read (of course, 1234
  29.     cannot be written 1234 in binary, the it is clear that
  30.     additionally we need a radix equal or greater than the
  31.     number of objects to be permuted in order to perform the
  32.     present experiment).
  33.      
  34.     Well, first the target: Permutations without repetitions
  35.     in the naturals.
  36.      
  37.     1) Pick 2 different permutations without repetitions
  38.     for the first 5 non-negative integers, or like I've
  39.     chosen to define and call it (please see also:
  40.     http://pastebin.com/9LNqD8ka):
  41.      
  42.     Please pick two different natural permutations for radix 5.
  43.      
  44.     Let's say 01234 and 30421.
  45.      
  46.     2) Enable the "radix or base-N" mode in your calculator,
  47.     set it to decimal and write once the following:
  48.      
  49.     -------------------------------------------
  50.     (30421-01234)/(10-1)
  51.      
  52.      
  53.     -------------------------------------------
  54.      
  55.     When ready press "equal" (=), and see the result,
  56.     (it usually should appears below at the right side.)
  57.      
  58.     -------------------------------------------
  59.     (30421-01234)/(10-1)
  60.                                            Dec
  61.                                           3243
  62.     -------------------------------------------
  63.      
  64.     Now without deleting the expression, change
  65.     with just one strike to hexadecimal, and
  66.     observe the change in the answer field:
  67.      
  68.     -------------------------------------------
  69.     (30421-01234)/(10-1)
  70.                                            HEX
  71.                                       00000CAB
  72.     -------------------------------------------
  73.      
  74.     Now without doing other things, press again
  75.     "equal" (=). If done correctly, you will see:
  76.      
  77.     -------------------------------------------
  78.     (30421-01234)/(10-1)
  79.                                            HEX
  80.                                           3243
  81.     -------------------------------------------
  82.      
  83.     Well, don't believe me, just try it, turn
  84.     back the calculator radix to decimal, again
  85.     without deleting the expression, and only
  86.     hitting one time the proper key. Look once
  87.     the result:
  88.      
  89.     -------------------------------------------
  90.     (30421-01234)/(10-1)
  91.                                             Dec
  92.                                           12867
  93.     -------------------------------------------
  94.      
  95.     Don't be worried. Press "equals" (=)
  96.      
  97.     -------------------------------------------
  98.     (30421-01234)/(10-1)
  99.                                             Dec
  100.                                            3243
  101.     -------------------------------------------
  102.      
  103.     Arrived this point. Are you intrigued?...
  104.     perhaps you might be so. If you now scroll
  105.     up to the beginning text for this exercise
  106.     here in this window or panel, you will notice
  107.     that this result is identical to the other
  108.     corresponding to the first time you stroked
  109.     "equal" (=). Sure, check it.
  110.      
  111.     yes is it!
  112.      
  113.     For (A<B) any pair of distinct permutations
  114.     without repetitions writable in your calculator,
  115.     and the expression (B-A)/(10-1) evaluated in all
  116.     the radices supported by your device (such that
  117.     each radix is greater than the number of digits
  118.     in your permutation), hitting "equal" (=) each
  119.     time you set a different radix you will find
  120.     that, in such base the resulting quotient is
  121.     written always using the same "digits" or
  122.     numerals. This and other properties are
  123.     already documented (or the author tried to
  124.     do this) as the sequences A215940,
  125.     A217626 and A196020, inside the
  126.     OEIS(TM) database (a.k.a Sloane's encyclopedia).
  127.      
  128.     Now the second part of this experiment:
  129.      
  130.     Try all the procedure described before but
  131.     now for permutations with repetitions...
  132.      
  133.     Bingo: the "it doesn't behave in the same way",
  134.     in fact The permutations with repetitions
  135.     like 12343 and 12334 doesn't have this
  136.     nice property.
  137.  
  138. *********************************************
  139.    Essential corrections:
  140. *********************************************
  141.  
  142. Let me advice about this... Counter-example:
  143.  
  144. Two permutations with repetitions destroy my
  145. hypothesis: (3114 - 1314)\(10 - 1), the resulting
  146. quotient always is written as "200".
  147.  
  148. **Then, what is happening there??: Simple,
  149. the right explanation is, for being invariant,
  150. the studied quotients should not contain one of
  151. the digits that have assigned (10-1) for some
  152. radix under consideration. Example: In decimal
  153. the quotient (3114 - 1341)\(10 - 1) looks like
  154. "197", then, neither octal nor decimal should
  155. be included inside the invariance set of radices,
  156. since (10-1)= 9 in decimal and (10-1)= 7 in octal.
  157.  
  158. This is in fact and additional restriction I didn't
  159. considered before (to Jan 7 2012 3:01 GMT).
  160.  
  161. The invariance property is satisfied only for
  162. those set of radices such that the considered
  163. quotients doesn't contain the symbol for (10-1).
  164.  
  165. But it is the final result of sparse distribution
  166. of time dedicated to analyze and explain these
  167. matters. So we need to talk about invariance,
  168. but defining precisely two sets: The quotients
  169. claimed to be invariants under radix reading
  170. interpretation, and the radices where they
  171. are invariants.
  172.  
  173. Final comment: A215940 was always worked by me
  174. mainly in decimal. Until tried A217626 for 11!
  175. (base-11) permutations, where arises for
  176. "first time" the symmetry breakdown and
  177. the negative terms, I couldn't be able
  178. to identify the last restriction. But
  179. this is worthy for studying in the
  180. further from an algebraic viewpoint
  181. based on Diophantine series.
  182.  
  183. There it will be obvious that being:
  184.  
  185. f_{i}(N)=sum_{k=0..(N-1)}P(i,k)*10^k
  186.  
  187. A permutation, if some P(i,u) were (10-1) then:
  188.  
  189. P(i,k=u)*10^k= (10-1)*10^u= -1*10^u + 10^(u+1),
  190.  
  191. Clearly it adds a unit to another "digit"... and
  192. subtract its corresponding power. This would be
  193. what destroys the expected pattern of invariance.
  194.  
  195. Please also try this: (21-12)\(10-1), and after
  196. only (21-12). The quotient always is 1, but the
  197. difference is distinct for each radix:
  198.  
  199. Is just (10-1) in the studied radix.
  200.  
  201. This accounts for this final correction in the
  202. explanation and interpretation given for the
  203. present experiment.
  204.  
  205. Also historically (at least for me), upon the need
  206. of learn how to compute generalized determinant
  207. polynomials, the observation that in decimal 12+9=21
  208. was the beginning of all the research.
  209.  
  210. Then for the regarded radices, each possible (10-1)
  211. cannot be PART of one of the quotients claimed
  212. to be invariants.
  213.  
  214. End-of-the-experiment.
  215.    
  216. Regards,
  217.      
  218. R. J. Cano, <aallggoorriitthhmmuuss@gmail.com>
  219. Jan 7th 2013, 11:45am (VET)
  220.  
  221. Acknowledgements:
  222. Thanks to OEIS(TM) for the patience, the support and
  223. for being all of you good examples and source of
  224. inspiration for get the best for our sequences.
RAW Paste Data