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
Keywords
Examples
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}}
Links
- Antti Karttunen, Table of n, a(n) for n = 1..1430
- Antti Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence).
- Index entries for sequences that are permutations of the natural numbers
Crossrefs
Programs
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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;
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Mathematica
bracket[ tree_ ] := (Flatten[ {tree, 0} ]/. 0->{0})//.{1, z___, 1, a_List, b_List, y___}:>{1, z, {1, a, b}, y}; widthfirst[ dectree_ ] := b2d[ Drop[ Flatten[ {Table[ Cases[ Level[ #, {k}, z ], A014486%20*)%20Ordering%5BReverse%5Bwidthfirst%20/@%20b2d%20/@%20wood%5B6%5D%5D%5D%20(*%20_Wouter%20Meeussen">Integer ], {k, Depth[ # ]-1} ] }/.z->List ], -1 ] ] & @(bracket@d2b[ dectree ]); (* uses functions in A014486 *) Ordering[Reverse[widthfirst /@ b2d /@ wood[6]]] (* _Wouter Meeussen, Aug 19 2025 *)
Extensions
Additional comments from Antti Karttunen, Aug 11 2000
Comments