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A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).

%I A373557 #18 Jun 25 2024 05:24:58
%S A373557 2,3,4,4,3,5,5,3,5,4,6,6,3,6,4,6,5,6,5,4,7,7,3,7,4,7,5,7,5,4,7,6,7,6,
%T A373557 4,7,6,5,8,8,3,8,4,8,5,8,5,4,8,6,8,6,4,8,6,5,8,7,8,7,4,8,7,5,8,7,6,8,
%U A373557 7,6,5,9,9,3,9,4,9,5,9,5,4,9,6,9,6,4,9,6,5
%N A373557 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).
%C A373557 A strong Schreier set is a subset of the positive integers with cardinality less than the minimum element in the set (see Chu link).
%C A373557 Each term k of 2*A371176 can be put into a one-to-one correspondence with a strong Schreier set by interpreting the 1-based position of the ones in the binary expansion of k (where position 1 corresponds to the least significant bit) as the elements of the corresponding strong Schreier set.
%C A373557 Arranging the elements in each set in decreasing order results in the sets being listed in lexicographical order (see example). Cf. A373579 for the elements arranged in increasing order.
%C A373557 The number of sets having maximum element m is A000045(m-1).
%H A373557 Paolo Xausa, <a href="/A373557/b373557.txt">Table of n, a(n) for n = 1..10000</a> (rows 1..2261 of the triangle, flattened).
%H A373557 Alistair Bird, <a href="https://outofthenormmaths.wordpress.com/2012/05/13/jozef-schreier-schreier-sets-and-the-fibonacci-sequence/">Jozef Schreier, Schreier sets and the Fibonacci sequence</a>, Out Of The Norm blog, May 13 2012.
%H A373557 Hùng Việt Chu, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Chu2/chu9.pdf">The Fibonacci Sequence and Schreier-Zeckendorf Sets</a>, Journal of Integer Sequences, Vol. 22 (2019), Article 19.6.5.
%F A373557 T(n,k) = A373345(n,k) + 1.
%e A373557 Triangle begins:
%e A373557                                         Corresponding
%e A373557    n  A371176(2*n)  bin(A371176(2*n))   strong Schreier set
%e A373557                                         (this sequence)
%e A373557   ---------------------------------------------------------
%e A373557    1        2               10          {2}
%e A373557    2        4              100          {3}
%e A373557    3        8             1000          {4}
%e A373557    4       12             1100          {4, 3}
%e A373557    5       16            10000          {5}       Sets are
%e A373557    6       20            10100          {5, 3}    lexicographically
%e A373557    7       24            11000          {5, 4}    ordered
%e A373557    8       32           100000          {6}
%e A373557    9       36           100100          {6, 3}
%e A373557   10       40           101000          {6, 4}
%e A373557   11       48           110000          {6, 5}
%e A373557   12       56           111000          {6, 5, 4}
%e A373557   ...
%t A373557 Join[{{2}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[4, 400, 4], DigitCount[#, 2, 1] < IntegerExponent[#, 2] + 1 &]]]
%Y A373557 Subsequence of A373345.
%Y A373557 Cf. A000045, A007895 (conjectured row lengths), A371176, A373556, A373579, A373853 (row sums).
%K A373557 nonn,tabf,base,easy
%O A373557 1,1
%A A373557 _Paolo Xausa_, Jun 09 2024