A209862 - OEIS

%I #26 Sep 15 2021 09:51:45

%S 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 28,23,27,29,30,31,32,33,34,36,40,48,35,37,41,49,38,42,50,44,52,56,39,

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

%C 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 From _Gus Wiseman_, Aug 24 2021: (Start)

%C 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    1

%C    2  3

%C    4  5  6  7

%C    8  9 10 12 11 13 14 15

%C   16 17 18 20 24 19 21 25 22 26 28 23 27 29 30 31

%C Mathematica: Table[Total[2^(Append[#,n]-1)]&/@Subsets[Range[n-1]],{n,5}]

%C Row lengths are A000079 (shifted right). Also Column k = 1.

%C Row sums are A010036.

%C Using reverse-lexicographic order gives A059893.

%C Using lexicographic order gives A059894.

%C Taking binary indices to prime indices gives A339195 (or A019565).

%C The ordering of sets is A344084.

%C A version using Heinz numbers is A344085.

%C (End)

%H Antti Karttunen, <a href="/A209862/b209862.txt">Table of n, a(n) for n = 0..32767</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F a(n) = A209859(A036044(A209641(n))) = A209859(A056539(A209641(n))).

%e From _Gus Wiseman_, Aug 24 2021: (Start)

%e The terms, their binary expansions, and their binary indices begin:

%e    0:      ~ {}

%e    1:    1 ~ {1}

%e    2:   10 ~ {2}

%e    3:   11 ~ {1,2}

%e    4:  100 ~ {3}

%e    5:  101 ~ {1,3}

%e    6:  110 ~ {2,3}

%e    7:  111 ~ {1,2,3}

%e    8: 1000 ~ {4}

%e    9: 1001 ~ {1,4}

%e   10: 1010 ~ {2,4}

%e   12: 1100 ~ {3,4}

%e   11: 1011 ~ {1,2,4}

%e   13: 1101 ~ {1,3,4}

%e   14: 1110 ~ {2,3,4}

%e   15: 1111 ~ {1,2,3,4}

%e (End)

%Y Inverse permutation: A209861. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

%Y Cf. A010036, A026793, A048793, A111059, A147655, A246688, A246867, A261144, A272020, A294648, A339360.

%K nonn

%O 0,3

%A _Antti Karttunen_, Mar 24 2012