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Irregular triangle read by rows where row n lists (in decreasing order) the elements of the strong Schreier set encoded by A371176(2*n).
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%I #18 Jun 25 2024 05:24:58

%S 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 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 7,6,5,9,9,3,9,4,9,5,9,5,4,9,6,9,6,4,9,6,5

%N 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 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 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 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 The number of sets having maximum element m is A000045(m-1).

%H 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 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 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 T(n,k) = A373345(n,k) + 1.

%e Triangle begins:

%e Corresponding

%e n A371176(2*n) bin(A371176(2*n)) strong Schreier set

%e (this sequence)

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

%e 1 2 10 {2}

%e 2 4 100 {3}

%e 3 8 1000 {4}

%e 4 12 1100 {4, 3}

%e 5 16 10000 {5} Sets are

%e 6 20 10100 {5, 3} lexicographically

%e 7 24 11000 {5, 4} ordered

%e 8 32 100000 {6}

%e 9 36 100100 {6, 3}

%e 10 40 101000 {6, 4}

%e 11 48 110000 {6, 5}

%e 12 56 111000 {6, 5, 4}

%e ...

%t Join[{{2}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[4, 400, 4], DigitCount[#, 2, 1] < IntegerExponent[#, 2] + 1 &]]]

%Y Subsequence of A373345.

%Y Cf. A000045, A007895 (conjectured row lengths), A371176, A373556, A373579, A373853 (row sums).

%K nonn,tabf,base,easy

%O 1,1

%A _Paolo Xausa_, Jun 09 2024