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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.*,

- b1=vecsubs( A217626,[1,9,2,9,1],[A1])
- b1=vecsubs( A0,[2, 18, 4, 18, 2],[A0]) \\ this (= 2*A1) should not be called A0
- b2=vecsubs( b1,[1, 19, 3, 8, 2],[A2a])
- b2=vecsubs( b2,[2, 8, 3, 19, 1],[A2b]) \\ the (..a, ..b) are pairs of "reversed" patterns
- b3=vecsubs( b2 ,[1, 29, 4, 7, 3],[A3a])
- b3=vecsubs( b3 ,[3, 7, 4, 29, 1],[A3b])
- b4=vecsubs( b3,[1, 39, 5, 6, 4],[A4a])
- b4=vecsubs( b4,[4, 6, 5, 39, 1],[A4b])
- b5=vecsubs( b4,[2, 28, 5, 17, 3],[A5a])
- b5=vecsubs( b5,[3, 17, 5, 28, 2],[A5b])
- 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 ...