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A131271 Triangular array T(n,k), n>=0, k=1..2^n, read by rows in bracketed pairs such that highest ranked element is bracketed with lowest ranked.

%I A131271 #15 Jun 22 2025 04:08:12
%S A131271 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,
%T A131271 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,
%U A131271 30,14,19,6,27,11,22,1,64,32,33,16,49,17,48,8,57
%N A131271 Triangular array T(n,k), n>=0, k=1..2^n, read by rows in bracketed pairs such that highest ranked element is bracketed with lowest ranked.
%C A131271 In a knockout competition with 2^n players, arranging the competition brackets (see Wikipedia) in T(n,k) order, where T(n,k) is the rank of the k-th 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. - _Colin Hall_, Feb 28 2012
%C A131271 Ranks in natural order of 2^n increasing real numbers appearing in limit cycles of interval iterations, or Median Spiral Order. [See the works by Demongeot & Waku]
%C A131271 From _Andrey Zabolotskiy_, Dec 06 2024 (Start):
%C A131271 For n>0, row n-1 is the permutation relating row n of Kepler's tree of fractions with row n of the left half of Stern-Brocot tree. Specifically, if K_n(k) [resp. SB_n(k)] is the k-th fraction in the n-th row of A294442 [resp. A057432], where 1/2 is in row 1 and k=1..2^(n-1), then K_n(k) = SB_n(T(n-1, k)).
%C A131271 The inverse permutation is row n of A088208.
%C A131271 When 1 is subtracted from each term, rows 3-5 become A240908, A240909, A240910. (End)
%H A131271 Alois P. Heinz, <a href="/A131271/b131271.txt">Table of n, a(n) for n = 0..8190</a>
%H A131271 John M. Campbell, <a href="https://arxiv.org/abs/2506.14072">A generalization of Deterministic Finite Automata related to discharging</a>, arXiv:2506.14072 [cs.FL], 2025. See p. 7.
%H A131271 Jacques Demongeot and Jules Waku, <a href="https://doi.org/10.1080/08898480500301785">Counter-Examples about Lower- and Upper-Bounded Population Growth</a>, Math. Pop. Studies 12 (2005), 199-210.
%H A131271 Jacques Demongeot and Jules Waku, <a href="https://doi.org/10.1098/rsta.2009.0177">Application of interval iterations to the entrainment problem in respiratory physiology</a>, Phil. Trans. R. Soc. A, 367 (2009), 4717-4739.
%H A131271 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bracket_%28tournament%29">Bracket (tournament)</a>
%F A131271 T(0,1) = 1, T(n,2k-1) = T(n-1,k), T(n,2k) = 2^n+1 - T(n-1,k).
%F A131271 T(n,1) = 1; for 1 < k <= 2^n, T(n,k) = 1 + (2^n)/m - T(n,k-m), where m = A006519(k-1). - _Mathew Englander_, Jun 20 2021
%e A131271 Triangle begins:
%e A131271 1;
%e A131271 1,  2;
%e A131271 1,  4, 2, 3;
%e A131271 1,  8, 4, 5, 2,  7, 3,  6;
%e A131271 1, 16, 8, 9, 4, 13, 5, 12, 2, 15, 7, 10, 3, 14, 6, 11;
%e A131271 ...
%p A131271 T:= proc(n,k) option remember;
%p A131271       `if`({n, k} = {1}, 1,
%p A131271       `if`(irem(k, 2)=1, T(n-1, (k+1)/2), 2^(n-1)+1 -T(n-1, k/2)))
%p A131271     end:
%p A131271 seq(seq(T(n, k), k=1..2^(n-1)), n=1..7); # _Alois P. Heinz_, Feb 28 2012, with offset 1
%t A131271 T[0, 1] = 1;
%t A131271 T[n_, k_] := T[n, k] = If[Mod[k, 2] == 1, T[n, (k + 1)/2], 2^n + 1 - T[n, k/2]];
%t A131271 Table[T[n, k], {n, 0, 6}, {k, 2^n}] // Flatten (* _Jean-François Alcover_, May 31 2018, after _Alois P. Heinz_ *)
%Y A131271 Cf. A049773, A088208, A294442, A057432.
%Y A131271 Cf. A005578 (last elements in rows), A155944 (T(n,2^(n-1)) for n>0).
%Y A131271 Cf. A006519, A240908, A240909, A240910.
%K A131271 nonn,tabf
%O A131271 0,3
%A A131271 J. Demongeot (Jacques.Demongeot(AT)imag.fr), Jun 24 2007
%E A131271 Edited (with new name from _Colin Hall_) by _Andrey Zabolotskiy_, Dec 06 2024