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User talk:M. F. Hasler/A217626

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The essential "distillate" of what I can say about this is put as comment in the OEIS entry A217626:

First differences of A215940, or first differences of permutations of 123...m, divided by 9 (with m sufficiently large, m!>n).

Especially some explicit formulae...

What is interesting, makes this "universal" and worth studying, is that any initial segment of arbitrary length of this sequence does not depend on m, provided that it is chosen large enough.

Example:

  • m=3, B=10: (123, 132, 213, 231, ...), D(%) = (9, 81, 18,...) = 9*(1, 9, 2,...) : here 9 = 10-1 = B-1
  • m=4, B=10: (1234,1243,1324,1342,...), D(%) = (9, 81, 18,...) = 9*(1, 9, 2,...) : here 9 = 10-1 = B-1

(A further investigation could be about the dependence on the base B: Of course, "divided by 9" would need to be read as "divided by 10-1", i.e., "divided by B-1": Example,

  • m=3, B=4: (123[4]=27, 132[4]=30, 213[4]=39, 231[4]=45,...), D(%) = (3,9,6,...) = 3*(1,3,2,...), here 3 = 4-1 = B-1)

To compactify the sequence, I used PARI to replace a given pattern with a label, e.g.,

  1. b1=vecsubs( A217626,[1,9,2,9,1],[A1])
  2. b1=vecsubs( A0,[2, 18, 4, 18, 2],[A0]) \\ this (= 2*A1) should not be called A0
  3. b2=vecsubs( b1,[1, 19, 3, 8, 2],[A2a])
  4. b2=vecsubs( b2,[2, 8, 3, 19, 1],[A2b]) \\ the (..a, ..b) are pairs of "reversed" patterns
  5. b3=vecsubs( b2 ,[1, 29, 4, 7, 3],[A3a])
  6. b3=vecsubs( b3 ,[3, 7, 4, 29, 1],[A3b])
  7. b4=vecsubs( b3,[1, 39, 5, 6, 4],[A4a])
  8. b4=vecsubs( b4,[4, 6, 5, 39, 1],[A4b])
  9. b5=vecsubs( b4,[2, 28, 5, 17, 3],[A5a])
  10. b5=vecsubs( b5,[3, 17, 5, 28, 2],[A5b])
  11. c1=vecsubs( b5,[A1, 78, A2a, 77, A2b, 78, A1],[B0])
(1) A1, 78, A2a, 77, A2b, 78, A1,  \\ NB: (1) = (1)'
(2)  657, A1, 178, A3a, 66, A0, 67, A2a, 646, A2a, 67, A3a,
(3)   176,     
(2)' A3b, 67, A2b, 646, A2b, 67, A0, 66, A3b, 178, A1, 657, \\ NB: (2)(3)(2)'  = [(2)(3)(2)']'
(1) A1, 78, A2a, 77, A2b, 78, A1,  \\ =  NB: (1)(2)(3)(2)'(1)  = [(1)(2)(3)(2)'(1)]'
(4)     5336, 
(1) A1, 78, A2a, 77, A2b, 78, A1, 
(5)      1657, A1, 278, A4a, 55, A5a, 56, A3a, 535, A2a, 167, A4a, 165, A5b, 56, A0, 535, 
(6)        A2b, 167, A5a, 55, A5b, 167, A2a, 
(7)         546, A1, 178, A3a, 66, A0, 67, A2a, 
(8)        5225, A1, 178, A3a, 66, A0, 67, A2a,   \\ could be called (7b) or (7'): differs only in 1 element
(9)         546, A1, 278, A4a, 55, A5a, 56, A3a, 1635, 
(a)          A3a, 56, A4a, 275, A4b, 56, A3b,   \\ NB : (a) = (a)'
(b)           525, A0, 56, A5a, 165, A4b, 167, 
(c)            A2b, 536, A2a,   \\ = (c)'
(d)             67, A3a, 176, A3b, 67,     \\ = (d)'
(e)              A2b, 5215, A2a,      \\ = (e)', could also be called (c2) or (c'), differs only in 1 element
(d)=(d)'        67, A3a, 176, A3b, 67, 
(c)=(c)'       A2b, 536, A2a, 
(b)'          167, A4a, 165, A5b, 56, A0, 525, 
(a)=(a')     A3a, 56, A4a, 275, A4b, 56, A3b, 
(9)'        1635, A3b, 56, A5b, 55, A4b, 278, A1, 546, 
(8)'       A2b, 67, A0, 66, A3b, 178, A1, 5225, 
(7)'        A2b, 67, A0, 66, A3b, 178, A1, 546, 
(6)=(6)'   A2b, 167, A5a, 55, A5b, 167, A2a, 
(5)'     535, A0, 56, A5a, 165, A4b, 167, A2b, 535, A3b, 56, A5b, 55, A4b, 278, A1, 1657, 
(1) A1, 78, A2a, 77, A2b, 78, A1, 
(4)     5336
...