cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-16 of 16 results.

A358524 Binary encoding of balanced ordered rooted trees (counted by A007059).

Original entry on oeis.org

0, 2, 10, 12, 42, 52, 56, 170, 204, 212, 232, 240, 682, 820, 844, 852, 920, 936, 976, 992, 2730, 3276, 3284, 3380, 3404, 3412, 3640, 3688, 3736, 3752, 3888, 3920, 4000, 4032, 10922, 13108, 13132, 13140, 13516, 13524, 13620, 13644, 13652, 14568, 14744, 14760
Offset: 1

Views

Author

Gus Wiseman, Nov 21 2022

Keywords

Comments

An ordered tree is balanced if all leaves are the same distance from the root.
The binary encoding of an ordered tree (see A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

Examples

			The terms together with their corresponding trees begin:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  204: ((o)(o))
  212: ((ooo))
  232: (((oo)))
  240: ((((o))))
  682: (ooooo)
  820: ((o)(oo))
  844: ((oo)(o))
  852: ((oooo))
  920: (((o)(o)))
  936: (((ooo)))
  976: ((((oo))))
  992: (((((o)))))

Crossrefs

These trees are counted by A007059.
This is a subset of A014486.
The version for binary trees is A057122.
The unordered version is A184155, counted by A048816.
Another ranking of balanced ordered trees is A358459.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A001263, A003238, A244925, A358377, A358379, A358523.

Programs

  • Mathematica
    binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]]
Select[Range[0,1000],binbalQ[#]&&SameQ@@Length/@Position[bint[#],{}]&]

A358553 Number of internal (non-leaf) nodes in the n-th standard ordered rooted tree.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 2, 3, 2, 2, 1, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 4, 3, 2, 5, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 2, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 3, 5, 4, 4, 3, 3, 2, 4, 5, 5, 4, 5, 4, 4, 3, 5, 3, 5, 4, 5, 4, 4
Offset: 1

Views

Author

Gus Wiseman, Nov 26 2022

Keywords

Comments

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

Examples

			The 89-th standard rooted tree is ((o)o(oo)), and it has 3 internal nodes, so a(89) = 3.

Crossrefs

This statistic is counted by A001263, unordered A358575 (reverse A055277).
The unordered version is A342507, firsts A358554.
Other statistics: A358371 (leaves), A358372 (nodes), A358379 (edge-height).
A000081 counts rooted trees, ordered A000108.
Cf. A000891, A014221, A358373, A358585, A358586, A358588.

Programs

  • Mathematica
    stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Table[Count[srt[n],[_],{0,Infinity}],{n,100}]

A358550 Depth of the ordered rooted tree with binary encoding A014486(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 3, 4, 2, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 2, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 3, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 2, 3, 3, 3, 4, 3, 3, 3, 3, 4, 4, 4, 4, 5, 3, 3, 3, 3, 4, 3, 3, 3
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2022

Keywords

Comments

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

Examples

			The first few rooted trees in binary encoding are:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))

Crossrefs

Positions of first appearances are A014137.
Leaves of the ordered tree are counted by A057514, standard A358371.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
Positions of 2's are A155587, indices of A020988.
The standard ranking of the ordered tree is A358523.
Nodes of the ordered tree are counted by A358551, standard A358372.
For standard instead of binary encoding we have A358379.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists all binary encodings.
Cf. A001263, A057122, A358373, A358505, A358524.

Programs

  • Mathematica
    binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[Depth[bint[k]]-1,{k,Select[Range[0,1000],binbalQ]}]

A358551 Number of nodes in the ordered rooted tree with binary encoding A014486(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7
Offset: 1

Views

Author

Gus Wiseman, Nov 22 2022

Keywords

Comments

The binary encoding of an ordered tree (A014486) is obtained by replacing the internal left and right brackets with 0's and 1's, thus forming a binary number.

Examples

			The first few rooted trees in binary encoding are:
    0: o
    2: (o)
   10: (oo)
   12: ((o))
   42: (ooo)
   44: (o(o))
   50: ((o)o)
   52: ((oo))
   56: (((o)))
  170: (oooo)
  172: (oo(o))
  178: (o(o)o)
  180: (o(oo))
  184: (o((o)))

Crossrefs

Run-lengths are A000108.
Binary encodings are listed by A014486.
Leaves of the ordered tree are counted by A057514, standard A358371.
Branches of the ordered tree are counted by A057515.
Edges of the ordered tree are counted by A072643.
The Matula-Goebel number of the ordered tree is A127301.
For standard instead of binary encoding we have A358372.
The standard ranking of the ordered tree is A358523.
Depth of the ordered tree is A358550, standard A358379.
Cf. A000081, A001263, A057122, A358373, A358505, A358524.

Programs

  • Mathematica
    binbalQ[n_]:=n==0||Count[IntegerDigits[n,2],0]==Count[IntegerDigits[n,2],1]&&And@@Table[Count[Take[IntegerDigits[n,2],k],0]<=Count[Take[IntegerDigits[n,2],k],1],{k,IntegerLength[n,2]}];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[Count[bint[k],_,{0,Infinity}],{k,Select[Range[0,10000],binbalQ]}]

Formula

a(n) = A072643(n) + 1.

A358459 Numbers k such that the k-th standard ordered rooted tree is balanced (counted by A007059).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 35, 37, 41, 43, 64, 128, 129, 137, 139, 163, 169, 171, 256, 257, 293, 512, 515, 529, 547, 553, 555, 641, 649, 651, 675, 681, 683, 1024, 1025, 2048, 2053, 2057, 2059, 2177, 2185, 2187, 2211, 2217, 2219, 2305, 2341, 2563
Offset: 1

Views

Author

Gus Wiseman, Nov 19 2022

Keywords

Comments

An ordered tree is balanced if all leaves have the same distance from the root.
We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

Examples

			The terms together with their corresponding ordered trees begin:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   8: (ooo)
   9: ((oo))
  11: ((o)(o))
  16: (oooo)
  17: ((((o))))
  32: (ooooo)
  35: ((oo)(o))
  37: (((o))((o)))
  41: ((o)(oo))
  43: ((o)(o)(o))

Links

Crossrefs

These trees are counted by A007059.
The unordered version is A184155, counted by A048816.
A000108 counts ordered rooted trees, unordered A000081.
A358379 gives depth of standard ordered trees.
Cf. A003238, A004249, A032027, A244925, A290822, A318185, A358373-A358378.

Programs

  • Mathematica
    stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Select[Range[100],SameQ@@Length/@Position[srt[#],{}]&]

A358525 Number of distinct permutations of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 3, 3, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 1, 2, 2, 3, 1, 6, 6, 4, 2, 6, 1, 6, 6, 6, 6, 5, 2, 3, 6, 4, 6, 6, 6, 5, 3, 4, 6, 5, 4, 5, 5, 1, 1, 2, 2, 3, 2, 6, 6, 4, 2, 3, 3, 12, 3, 12, 12, 5, 2, 6, 3, 12, 3, 4
Offset: 0

Views

Author

Gus Wiseman, Nov 21 2022

Keywords

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The a(45) = 6 permutations are: (2121), (2112), (2211), (1221), (1212), (1122).

Links

Crossrefs

See link for sequences related to standard compositions.
Positions of 1's are A272919.
Cf. A000120, A048896, A070939, A329395, A333766, A358371, A358372, A358379, A358505, A358507, A358508.

Programs

  • Mathematica
    stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Table[Length[Permutations[stc[n]]],{n,0,100}]
Previous Showing 11-16 of 16 results.

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