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Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).
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%I #45 Sep 13 2024 15:56:35

%S 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 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 6,5,4,8,8,2,8,3,8,4,8,4,3,8,5,8,5,3,8,5,4,8,6

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

%C A Schreier set is a subset of the positive integers with cardinality less than or equal to the minimum element in the set.

%C 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 See A373359 for the elements in each set arranged in increasing order.

%C The number of sets having maximum element m is A000045(m).

%H 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 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 T(n,k) = A373557(n,k) - 1.

%e Triangle begins:

%e Corresponding Schreier

%e n A371176(n) bin(A371176(n)) set (this sequence)

%e -------------------------------------------------------

%e 1 1 1 {1}

%e 2 2 10 {2}

%e 3 4 100 {3}

%e 4 6 110 {3, 2}

%e 5 8 1000 {4}

%e 6 10 1010 {4, 2}

%e 7 12 1100 {4, 3}

%e 8 16 10000 {5}

%e 9 18 10010 {5, 2}

%e 10 20 10100 {5, 3}

%e 11 24 11000 {5, 4}

%e 12 28 11100 {5, 4, 3}

%e ...

%t Join[{{1}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 200, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]]]

%Y Cf. A000045, A371176, A373359, A373556, A373557.

%Y Cf. A007895 (conjectured row lengths), A072649 (first column), A373346 (row sums), A373347.

%K nonn,tabf,easy,base

%O 1,2

%A _Paolo Xausa_, Jun 01 2024