A368751 Irregular triangle read by rows: T(n,k) is the number of co-atoms contained in the k-th balanced string of left/right parentheses of length 2*n, where strings within a row are in reverse lexicographical order.

1, 0, 1, 2, 1, 1, 0, 0, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 1, 2, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 2, 2, 3, 2, 3, 4, 3, 3, 2, 2, 2, 3, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 2, 1, 1, 1, 2, 1, 1, 0, 0, 1, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0

Offset

1,4

Comments

See A368750 for the definition of balanced strings and atoms/co-atoms.

References

Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, exercise 60, p. 478.

Links

Paolo Xausa, Table of n, a(n) for n = 1..17576 (rows 1..8 of the triangle, flattened).

Formula

T(n,k) = A368752(n,k) - A368750(n,k).

Example

Triangle begins:
  [1] 1 0;
  [2] 1 2 1 1 0 0;
  [3] 1 1 2 1 2 3 2 2 1 1 1 2 1 1 0 0 1 0 0 0;
  ...
The strings corresponding to row 2, in reverse lexicographical order, are:
  "))((" (1 co-atom),
  ")()(" (2 co-atoms),
  ")(()" (1 co-atom),
  "())(" (1 co-atom),
  "()()" (0 co-atoms) and
  "(())" (0 co-atoms).

Mathematica

strings[n_]:=Permutations[PadLeft[PadLeft[{}, n, 1], 2n, -1]];
Array[Map[SequenceCount[Accumulate[#], {-1, 0}]&, strings[#]]&, 5]

Crossrefs

Cf. A000346 (row sums), A000984 (row lengths), A362030 and A368804 (binary words).

Cf. A368750 (atoms), A368752 (all atoms), A368753 (defects).

Sequence in context: A284996 A215029 A362832 * A025908 A134404 A334134

Adjacent sequences: A368748 A368749 A368750 * A368752 A368753 A368754

Keyword

nonn,tabf

Author

Paolo Xausa, Jan 05 2024

Status

approved