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Search: id:A125644
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| A125644 |
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Table with T(n,k) = the number of Dyck paths whose ascent lengths, in order, are the k-th composition of n, for the standard composition order. |
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1
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| 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 3, 1, 2, 1, 1, 1, 4, 3, 6, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 1, 5, 4, 10, 3, 9, 6, 10, 2, 7, 5, 9, 3, 7, 4, 5, 1, 4, 3, 6, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 1, 6, 5, 15, 4, 14, 10, 20, 3, 12, 9, 19, 6, 16, 10, 15, 2, 9, 7, 16, 5, 14, 9, 14, 3, 10, 7, 12, 4, 9, 5, 6, 1, 5
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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The standard composition order is specified in A066099. Also, the number of ordered, rooted trees for which the out-degrees of the non-leaf nodes, in preorder, form the specified composition.
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FORMULA
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For a composition [b_1,...,b_m], we have a([b_1]) = 1, a([b_1,b_2,...,b_m]) = Sum_{i=0}^{b_1-1} a([b_2+i,b_3,...,b_m]).
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EXAMPLE
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Composition number 11 is 2,1,1; there are 3 Dyck paths with this pattern: UUDDUDUD, UUDUDDUD and UUDUDUDD, so a(11) = 3. The corresponding trees are:
....O ..... O....
....| ..... |....
....O O...O O....
....| |...| |....
O...O O...O O...O
.\./. .\./. .\./.
..O.. ..O.. ..O..
The table starts:
1,
1,
1,1,
1,2,1,1,
1,3,2,3,1,2,1,1,
1,4,3,6,2,5,3,4,1,3,2,3,1,2,1,1,
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CROSSREFS
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Cf. A125181, A011782 (row lengths), A000108 (row sums), A066099.
Sequence in context: A029332 A134431 A070879 this_sequence A048821 A120221 A094899
Adjacent sequences: A125641 A125642 A125643 this_sequence A125645 A125646 A125647
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KEYWORD
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nonn,tabf
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AUTHOR
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Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net), Nov 29 2006
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