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A358373
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Triangle read by rows where row n lists the sorted standard ordered rooted tree-numbers of all unlabeled ordered rooted trees with n vertices.
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17
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 33, 65, 129, 257, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 41, 49, 50, 57, 66, 97, 130, 193, 258, 385, 513, 514, 769, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073
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OFFSET
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1,2
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COMMENTS
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We define the standard ordered rooted tree (SORT)-number of an unlabeled ordered rooted tree to be one plus the standard composition number (A066099) of the SORT-numbers of the branches, or 1 if there are no branches. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.
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LINKS
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EXAMPLE
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Triangle begins:
1
2
3 4
5 6 7 8 9
10 11 12 13 14 15 16 17 18 25 33 65 129 257
For example, the tree t = ((o,o),o) has branches (o,o) and o with SORT-numbers 4 and 1, and the standard composition number of (4,1) is 17, so t has SORT-number 18 and is found in row 5.
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MATHEMATICA
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stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
aotrank[t_]:=If[t=={}, 1, 1+stcinv[aotrank/@t]];
aot[n_]:=If[n==1, {{}}, Join@@Table[Tuples[aot/@c], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Sort[aotrank/@aot[n]], {n, 6}]
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CROSSREFS
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The version for compositions is A000027.
The unordered version (using Matula-Goebel numbers) is A061773.
The version for Heinz numbers of partitions is A215366.
The row containing n is A358372(n).
A001263 counts unlabeled ordered rooted trees by nodes and leaves.
A358371 counts leaves in standard ordered rooted trees.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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