OFFSET
0,3
COMMENTS
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.
From Gus Wiseman, Aug 24 2021: (Start)
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:
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 28 23 27 29 30 31
Mathematica: Table[Total[2^(Append[#,n]-1)]&/@Subsets[Range[n-1]],{n,5}]
Row lengths are A000079 (shifted right). Also Column k = 1.
Row sums are A010036.
Using reverse-lexicographic order gives A059893.
Using lexicographic order gives A059894.
The ordering of sets is A344084.
A version using Heinz numbers is A344085.
(End)
LINKS
EXAMPLE
From Gus Wiseman, Aug 24 2021: (Start)
The terms, their binary expansions, and their binary indices begin:
0: ~ {}
1: 1 ~ {1}
2: 10 ~ {2}
3: 11 ~ {1,2}
4: 100 ~ {3}
5: 101 ~ {1,3}
6: 110 ~ {2,3}
7: 111 ~ {1,2,3}
8: 1000 ~ {4}
9: 1001 ~ {1,4}
10: 1010 ~ {2,4}
12: 1100 ~ {3,4}
11: 1011 ~ {1,2,4}
13: 1101 ~ {1,3,4}
14: 1110 ~ {2,3,4}
15: 1111 ~ {1,2,3,4}
(End)
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 24 2012
STATUS
approved