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A384989 Order of the permutation of [n] formed by a Josephus elimination variation: take 3, skip 1.

%I A384989 #11 Jun 19 2025 18:34:33
%S A384989 1,1,1,1,2,3,4,4,6,10,6,8,12,11,9,10,42,15,16,52,60,120,18,30,140,99,
%T A384989 95,28,90,660,30,28,30,30,546,336,48,420,24,765,1680,60,308,400,66,
%U A384989 462,418,4830,252,1430,468,49,42,180,1020,52,2310,264,1680,340,380
%N A384989 Order of the permutation of [n] formed by a Josephus elimination variation: take 3, skip 1.
%C A384989 The Josephus elimination begins with a circular list [n] from which successively take 3 elements and skip 1, and the permutation is the elements taken in the order they're taken.
%C A384989 The same effect can be had by leaving remaining elements at the end of a flat list of [n] and applying the "skip" as a move (rotate) of the element at position 3*i+4 to the end of the list, for successive i >= 0.
%C A384989 Take 3 and move 1 is a move every 4th element, but with the next 4 elements reckoned inclusive of the element which replaced the moved 1, and hence positions 3 apart.
%C A384989 A given element can be skipped or moved multiple times before reaching its final position.
%H A384989 Chuck Seggelin, <a href="/A384989/b384989.txt">Table of n, a(n) for n = 1..1000</a>
%e A384989 For n=11, the rotations to construct the permutation are
%e A384989     1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
%e A384989              \------------------------/  1st rotation
%e A384989     1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 4
%e A384989                       \---------------/  2nd rotation
%e A384989     1, 2, 3, 5, 6, 7, 9, 10, 11, 4, 8
%e A384989                                  \----/  3rd rotation
%e A384989     1, 2, 3, 5, 6, 7, 9, 10, 11, 8, 4
%e A384989 The 3rd rotate is an example of an element (4) which was previously rotated to the end, being rotated to the end again.
%e A384989 This final permutation has order a(11) = 6 (applying it 6 times reaches the identity permutation again).
%o A384989 (Python)
%o A384989 from sympy.combinatorics import Permutation
%o A384989 def move_fourth(seq):
%o A384989     for i in range(3,len(seq),3):
%o A384989         seq.append(seq.pop(i))
%o A384989     return seq
%o A384989 def a(n):
%o A384989     seq = list(range(n))
%o A384989     p = move_fourth(seq.copy())
%o A384989     return Permutation(p).order()
%Y A384989 Cf. A051732 (Josephus elimination permutation order), A384753 (take 2 skip 1 Josephus variation).
%K A384989 nonn
%O A384989 1,5
%A A384989 _Chuck Seggelin_, Jun 14 2025