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A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).

%I A209862 #26 Sep 15 2021 09:51:45
%S A209862 0,1,2,3,4,5,6,7,8,9,10,12,11,13,14,15,16,17,18,20,24,19,21,25,22,26,
%T A209862 28,23,27,29,30,31,32,33,34,36,40,48,35,37,41,49,38,42,50,44,52,56,39,
%U A209862 43,51,45,53,57,46,54,58,60,47,55,59,61,62,63,64,65,66,68,72,80,96,67,69,73,81,97,70,74,82,98,76,84,100,88,104,112,71,75,83
%N A209862 Permutation of nonnegative integers which maps A209642 into ascending order (A209641).
%C A209862 Conjecture: For all n, a(A054429(n)) = A054429(a(n)), i.e. A054429 acts as a homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860.
%C A209862 From _Gus Wiseman_, Aug 24 2021: (Start)
%C A209862 As a triangle with row lengths 2^n, T(n,k) for n > 0 appears (verified up to n = 2^15) to be the unique nonnegative integer whose binary indices are the k-th subset of {1..n} containing n. Here, a binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion, and sets are sorted first by length, then lexicographically. For example, the triangle begins:
%C A209862    1
%C A209862    2  3
%C A209862    4  5  6  7
%C A209862    8  9 10 12 11 13 14 15
%C A209862   16 17 18 20 24 19 21 25 22 26 28 23 27 29 30 31
%C A209862 Mathematica: Table[Total[2^(Append[#,n]-1)]&/@Subsets[Range[n-1]],{n,5}]
%C A209862 Row lengths are A000079 (shifted right). Also Column k = 1.
%C A209862 Row sums are A010036.
%C A209862 Using reverse-lexicographic order gives A059893.
%C A209862 Using lexicographic order gives A059894.
%C A209862 Taking binary indices to prime indices gives A339195 (or A019565).
%C A209862 The ordering of sets is A344084.
%C A209862 A version using Heinz numbers is A344085.
%C A209862 (End)
%H A209862 Antti Karttunen, <a href="/A209862/b209862.txt">Table of n, a(n) for n = 0..32767</a>
%H A209862 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A209862 a(n) = A209859(A036044(A209641(n))) = A209859(A056539(A209641(n))).
%e A209862 From _Gus Wiseman_, Aug 24 2021: (Start)
%e A209862 The terms, their binary expansions, and their binary indices begin:
%e A209862    0:      ~ {}
%e A209862    1:    1 ~ {1}
%e A209862    2:   10 ~ {2}
%e A209862    3:   11 ~ {1,2}
%e A209862    4:  100 ~ {3}
%e A209862    5:  101 ~ {1,3}
%e A209862    6:  110 ~ {2,3}
%e A209862    7:  111 ~ {1,2,3}
%e A209862    8: 1000 ~ {4}
%e A209862    9: 1001 ~ {1,4}
%e A209862   10: 1010 ~ {2,4}
%e A209862   12: 1100 ~ {3,4}
%e A209862   11: 1011 ~ {1,2,4}
%e A209862   13: 1101 ~ {1,3,4}
%e A209862   14: 1110 ~ {2,3,4}
%e A209862   15: 1111 ~ {1,2,3,4}
%e A209862 (End)
%Y A209862 Inverse permutation: A209861. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.
%Y A209862 Cf. A010036, A026793, A048793, A111059, A147655, A246688, A246867, A261144, A272020, A294648, A339360.
%K A209862 nonn
%O A209862 0,3
%A A209862 _Antti Karttunen_, Mar 24 2012