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A208569 Triangular array T(n,k), n>=1, k=1..2^(n-1), read by rows in bracketed pairs such that highest ranked element is bracketed with lowest ranked. 2
1, 1, 2, 1, 4, 2, 3, 1, 8, 4, 5, 2, 7, 3, 6, 1, 16, 8, 9, 4, 13, 5, 12, 2, 15, 7, 10, 3, 14, 6, 11, 1, 32, 16, 17, 8, 25, 9, 24, 4, 29, 13, 20, 5, 28, 12, 21, 2, 31, 15, 18, 7, 26, 10, 23, 3, 30, 14, 19, 6, 27, 11, 22, 1, 64, 32, 33, 16, 49, 17, 48, 8, 57 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

In a knockout competition with m players, arranging the competition brackets (see links) in T(m,k) order, where T(m,k) is the rank of the player, ensures that highest ranked players cannot meet until the later stages of the competition.  None of the top 2^p ranked players can meet earlier than the p-th from last round of the competition.  At the same time the top ranked players in each match meet the lowest ranked player possible consistent with this rule.  The sequence for the top ranked players meeting the highest ranked player possible is A049773.

LINKS

Alois P. Heinz, Rows n = 1..13, flattened

Wikipedia, Bracket (tournament)

FORMULA

T(1,1) = 1, T(n,2k-1) = T(n-1,k), T(n,2k) = 2^(n-1)+1 - T(n-1,k).

EXAMPLE

Triangle begins:

1;

1,  2;

1,  4, 2, 3;

1,  8, 4, 5, 2,  7, 3,  6;

1, 16, 8, 9, 4, 13, 5, 12, 2, 15, 7, 10, 3, 14, 6, 11;

MAPLE

T:= proc(n, k) option remember;

      `if`({n, k} = {1}, 1,

      `if`(irem(k, 2)=1, T(n-1, (k+1)/2), 2^(n-1)+1 -T(n-1, k/2)))

    end:

seq(seq(T(n, k), k=1..2^(n-1)), n=1..7); # Alois P. Heinz, Feb 28 2012

CROSSREFS

Cf. A049773.  From row 4 onwards same as finite sequence A131271.

Cf. A005578 (last elements in rows), A155944 (T(n,2^(n-2)) for n>1).

Sequence in context: A123755 A118291 A118290 * A132223 A224712 A135941

Adjacent sequences:  A208566 A208567 A208568 * A208570 A208571 A208572

KEYWORD

nonn,tabf

AUTHOR

Colin Hall, Feb 28 2012

STATUS

approved

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Last modified August 19 08:57 EDT 2017. Contains 290794 sequences.