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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).
8
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, 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, 6, 5, 4, 8, 8, 2, 8, 3, 8, 4, 8, 4, 3, 8, 5, 8, 5, 3, 8, 5, 4, 8, 6
OFFSET
1,2
COMMENTS
A Schreier set is a subset of the positive integers with cardinality less than or equal to the minimum element in the set.
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).
See A373359 for the elements in each set arranged in increasing order.
The number of sets having maximum element m is A000045(m).
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.
FORMULA
T(n,k) = A373557(n,k) - 1.
EXAMPLE
Triangle begins:
Corresponding Schreier
n A371176(n) bin(A371176(n)) set (this sequence)
-------------------------------------------------------
1 1 1 {1}
2 2 10 {2}
3 4 100 {3}
4 6 110 {3, 2}
5 8 1000 {4}
6 10 1010 {4, 2}
7 12 1100 {4, 3}
8 16 10000 {5}
9 18 10010 {5, 2}
10 20 10100 {5, 3}
11 24 11000 {5, 4}
12 28 11100 {5, 4, 3}
...
MATHEMATICA
Join[{{1}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 200, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]]]
CROSSREFS
Cf. A007895 (conjectured row lengths), A072649 (first column), A373346 (row sums), A373347.
Sequence in context: A139169 A238444 A076742 * A376698 A308284 A036465
KEYWORD
nonn,tabf,easy,base
AUTHOR
Paolo Xausa, Jun 01 2024
STATUS
approved