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A368753
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Irregular triangle read by rows: T(n,k) is the defect of the k-th balanced string of left/right parentheses of length 2*n, where strings within a row are in reverse lexicographical order.
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5
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1, 0, 2, 2, 1, 1, 0, 0, 3, 3, 3, 2, 3, 3, 2, 2, 1, 1, 2, 2, 1, 1, 0, 0, 1, 0, 0, 0, 4, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 3, 3, 2, 2, 4, 4, 4, 3, 4, 4, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 2, 1, 1, 1, 3, 3, 3, 2, 3, 3, 2, 2, 1, 1, 2, 2, 1, 1, 0, 0, 1, 0, 0, 0, 2, 2, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0
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OFFSET
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1,3
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COMMENTS
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See A368750 for the definition of balanced strings and atoms/co-atoms.
The defect is half the length of co-atoms or, equivalently, the number of indices where the i-th right parenthesis precedes the i-th left parenthesis (see Knuth, 2011).
Knuth reports a result by MacMahon (1909) and Chung and Feller (1949): exactly A000108(n) balanced strings of length 2*n have defect d, for 0 <= d <= n.
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REFERENCES
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Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, exercise 60, pp. 478 and 797.
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LINKS
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EXAMPLE
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Triangle begins:
[1] 1 0;
[2] 2 2 1 1 0 0;
[3] 3 3 3 2 3 3 2 2 1 1 2 2 1 1 0 0 1 0 0 0;
...
The strings corresponding to row 2, in reverse lexicographical order, are:
"))((" (defect 2),
")()(" (defect 2),
")(()" (defect 1),
"())(" (defect 1),
"()()" (defect 0) and
"(())" (defect 0).
For the string "())((())))(()(", for example, the defect is calculated as follows:
.
atom
| co-atom
| | atom co-atom
| | | | co-atom
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() )( (()) ))(( )(
* ** *
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defect = length of co-atoms / 2 = 8 / 2 = 4 = number of indices where the i-th right parenthesis precedes the i-th left parenthesis (marked with asterisks).
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MATHEMATICA
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strings[n_]:=Permutations[PadLeft[PadLeft[{}, n, 1], 2n]];
defect[s_]:=Count[Position[s, 1]-Position[s, 0], _?Positive, {2}];
Array[Map[defect, strings[#]]&, 5]
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CROSSREFS
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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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