A038776 The sequence a[1] to a[ cat[n] ] is the permutation that converts forest[n] of depth-first planar planted binary trees into breadth-first representation.

1, 2, 4, 5, 3, 9, 10, 13, 14, 12, 6, 7, 8, 11, 23, 24, 27, 28, 26, 36, 37, 41, 42, 40, 32, 33, 35, 39, 15, 16, 18, 19, 17, 22, 25, 20, 21, 34, 38, 29, 30, 31, 65, 66, 69, 70, 68, 78, 79, 83, 84, 82, 74, 75, 77, 81, 106, 107, 111, 112, 110, 125, 126, 131, 132, 130

Offset

1,2

Links

A. Karttunen, Table of n, a(n) for n = 1..1430

A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)

Index entries for sequences that are permutations of the natural numbers

Example

tree ( 1 (100) (10 ) ) becomes (1) (11)(00 0 ) thus (1 (1(100) 0) ) and is permuted from position 3 in forest[3] to position 5 by permutation {1,2,4,5,3}={{1},{2},{4,5,3}}

Maple

[seq(CatalanRank(inf, (btbf2df(binrev(CatalanUnrank(inf, j)), 0, 1)/2))+1, j=0..(binomial(2*inf, inf)/(inf+1))-1)]; (In practice, use a value like 6 instead of infinity).
btbf2df := proc(nn, i, r) local n, j, c, x, y, w; n := nn; if(0 = (n mod 2)) then RETURN(0); fi; c := i; for j from 1 to r do c := c + (n mod 2); n := floor(n/2); od; w := 2*c; c := 0; for j from 1 to (2*i) do c := c + (n mod 2); n := floor(n/2); od; x := btbf2df(n, c, (w-(j-1))); y := btbf2df(floor(n/2), c+(n mod 2), (w-(j))); RETURN((2^(binwidth(x)+binwidth(y))) + (x * (2^(binwidth(y)))) + y); end;
floor_log_2 := proc(n) local nn, i: nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi: nn := floor(nn/2); od: end:
binwidth := n -> (`if`((0 = n), 1, floor_log_2(n)+1));
binrev := proc(nn) local n, z; n := nn; z := 0; while(n <> 0) do z := 2*z + (n mod 2); n := floor(n/2); od; RETURN(z); end;

Crossrefs

Compare to the plot of A082364 and A072619.

Inverse of A070041. Cf. also A038774, A038775. If "expanded" produces A057117. Max cycle lengths: A057542.

Sequence in context: A060736 A097292 A269780 * A210863 A245816 A118461

Adjacent sequences: A038773 A038774 A038775 * A038777 A038778 A038779

Keyword

nonn

Author

Wouter Meeussen, May 04 2000

Extensions

Additional comments from Antti Karttunen, Aug 11 2000

Status

approved