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A208569
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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.
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3
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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T(1,1) = 1, T(n,2k-1) = T(n-1,k), T(n,2k) = 2^(n-1)+1 - T(n-1,k).
T(n,1) = 1; for 1 < k <= 2^(n-1), T(n,k) = 1 + (2^n)/(2*m) - T(n,k-m), where m = A006519(k-1). - Mathew Englander, Jun 20 2021
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EXAMPLE
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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;
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MAPLE
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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
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[Union @ {n, k} == {1}, 1, If[Mod[k, 2] == 1, T[n - 1, (k + 1)/2], 2^(n - 1) + 1 - T[n - 1, k/2]]];
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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