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A346036 Number of swaps needed to return the recursive transform T(k) = concatenate(reverse(k), len(k) + 1) to the tuple 1, 2, ..., n.

%I A346036 #27 Sep 25 2021 23:09:11
%S A346036 0,0,1,2,2,4,5,4,6,8,7,10,10,10,13,12,12,14,17,16,18,18,17,22,22,20,
%T A346036 25,24,24,28,29,24,26,32,31,34,32,32,35,38,36,40,35,40,40,40,39,44,46,
%U A346036 42,49,50,44,52,51,52,52,54,51,54,58,56,59,54,54,64,61,60
%N A346036 Number of swaps needed to return the recursive transform T(k) = concatenate(reverse(k), len(k) + 1) to the tuple 1, 2, ..., n.
%e A346036 For n=7, the T transforms give tuple 6,4,2,1,3,5,7 (triangle A220073 row 7) which requires a(7) = 5 swaps to return to 1,2,3,4,5,6,7.
%o A346036 (Python)
%o A346036 def swaps(l, start = 0):
%o A346036   if len(l) == 0:
%o A346036     return 0
%o A346036   n = start + 1
%o A346036   if l[0] != n:
%o A346036     for i, x in enumerate(l):
%o A346036       if x == n:
%o A346036         l[0], l[i] = l[i], l[0]
%o A346036         return 1 + swaps(l[1:], n)
%o A346036     else:
%o A346036       raise
%o A346036   else:
%o A346036     return swaps(l[1:], n)
%o A346036 def T(l):
%o A346036   n = len(l)
%o A346036   l.reverse()
%o A346036   l.append(n + 1)
%o A346036   return l
%o A346036 if __name__ == "__main__":
%o A346036   l = [ ]
%o A346036   for n in range(1, 100):
%o A346036     l = T(l)
%o A346036     n_swaps = swaps(l[:])
%o A346036     print("{}, ".format(n_swaps), end="")
%o A346036 (PARI) a(n) = my(h=n>>1); n - #permcycles(vectorsmall(n,i, abs(2*i-n) + (i<=h))); \\ _Kevin Ryde_, Jul 23 2021
%Y A346036 Cf. A220073.
%K A346036 nonn
%O A346036 1,4
%A A346036 _Emanuel Landeholm_, Jul 02 2021