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A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

%I A373345 #45 Sep 13 2024 15:56:35
%S A373345 1,2,3,3,2,4,4,2,4,3,5,5,2,5,3,5,4,5,4,3,6,6,2,6,3,6,4,6,4,3,6,5,6,5,
%T A373345 3,6,5,4,7,7,2,7,3,7,4,7,4,3,7,5,7,5,3,7,5,4,7,6,7,6,3,7,6,4,7,6,5,7,
%U A373345 6,5,4,8,8,2,8,3,8,4,8,4,3,8,5,8,5,3,8,5,4,8,6
%N A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).
%C A373345 A Schreier set is a subset of the positive integers with cardinality less than or equal to the minimum element in the set.
%C A373345 Each term k of A371176 can be put into a one-to-one correspondence with a 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 Schreier set (see A371176 and Bird link).
%C A373345 See A373359 for the elements in each set arranged in increasing order.
%C A373345 The number of sets having maximum element m is A000045(m).
%H A373345 Paolo Xausa, <a href="/A373345/b373345.txt">Table of n, a(n) for n = 1..10000</a> (rows 1..2261 of the triangle, flattened).
%H A373345 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.
%F A373345 T(n,k) = A373557(n,k) - 1.
%e A373345 Triangle begins:
%e A373345                                    Corresponding Schreier
%e A373345    n  A371176(n)  bin(A371176(n))  set (this sequence)
%e A373345   -------------------------------------------------------
%e A373345    1      1              1         {1}
%e A373345    2      2             10         {2}
%e A373345    3      4            100         {3}
%e A373345    4      6            110         {3, 2}
%e A373345    5      8           1000         {4}
%e A373345    6     10           1010         {4, 2}
%e A373345    7     12           1100         {4, 3}
%e A373345    8     16          10000         {5}
%e A373345    9     18          10010         {5, 2}
%e A373345   10     20          10100         {5, 3}
%e A373345   11     24          11000         {5, 4}
%e A373345   12     28          11100         {5, 4, 3}
%e A373345   ...
%t A373345 Join[{{1}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 200, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]]]
%Y A373345 Cf. A000045, A371176, A373359, A373556, A373557.
%Y A373345 Cf. A007895 (conjectured row lengths), A072649 (first column), A373346 (row sums), A373347.
%K A373345 nonn,tabf,easy,base
%O A373345 1,2
%A A373345 _Paolo Xausa_, Jun 01 2024