Revision History for A358523
(Underlined text is an addition;
strikethrough text is a deletion.)
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#6 by Michael De Vlieger at Mon Nov 21 09:38:25 EST 2022
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#5 by Gus Wiseman at Mon Nov 21 04:49:34 EST 2022
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#4 by Gus Wiseman at Mon Nov 21 04:49:16 EST 2022
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#3 by Gus Wiseman at Mon Nov 21 04:48:46 EST 2022
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| COMMENTS
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The Matula-Goebelbinary numberencoding of aan rootedordered tree (A014486) is the product of primes indexedobtained by the Matula-Goebel numbers ofreplacing the branches of itsinternal root, whichleft givesand aright bijectivebrackets correspondencewith between0's positiveand integers1's, thus andforming unlabeleda rootedbinary treesnumber.
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.
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| CROSSREFS
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Cf. A001263 tri_ort_lvs, A014221 stnum_path, A057122, A358371 sot_lvs, A358372 sot_vts, A358373 tri_sotnums_vts, A358379 sot_depth.
Cf. `A001263, A014221, `A057122, A358371, A358372, A358373, A358379.
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#2 by Gus Wiseman at Mon Nov 21 00:51:41 EST 2022
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| NAME
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allocatedStandard ordered tree numbers of ordered trees in order of fortheir Gusbinary Wisemanencodings (A014486).
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| DATA
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1, 2, 4, 3, 8, 7, 6, 9, 5, 16, 15, 14, 25, 13, 12, 11, 18, 129, 65, 10, 33, 257, 17, 32, 31, 30, 57, 29, 28, 27, 50, 385, 193, 26, 97, 769, 49, 24, 23, 22, 41, 21, 36, 35, 258, 32769, 16385, 130, 8193, 16777217, 4097, 20, 19, 66
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| OFFSET
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0,2
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| COMMENTS
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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.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
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.
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| LINKS
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Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
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| EXAMPLE
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The first six binary encodings are: 0, 2, 10, 12, 42, 44, and the corresponding trees have standard ranks: 1, 2, 4, 3, 8, 7.
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| MATHEMATICA
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stcinv[q_]:=Total[2^Accumulate[Reverse[q]]]/2;
srtinv[t_]:=If[t=={}, 1, stcinv[srtinv/@t]+1];
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[srtinv[bint[n]], {n, Select[Range[0, 100], binbalQ]}]
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| CROSSREFS
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A dual sequence is A358505.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists all binary encodings.
Cf. A001263 tri_ort_lvs, A014221 stnum_path, A057122, A358371 sot_lvs, A358372 sot_vts, A358373 tri_sotnums_vts, A358379 sot_depth.
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| KEYWORD
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allocated
nonn
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| AUTHOR
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Gus Wiseman, Nov 21 2022
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| STATUS
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approved
editing
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#1 by Gus Wiseman at Sun Nov 20 11:31:36 EST 2022
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| NAME
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allocated for Gus Wiseman
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| KEYWORD
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allocated
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| STATUS
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approved
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