A368752 Irregular triangle read by rows: T(n,k) is the number of atoms + 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, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 3, 3, 3, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 4, 4, 3, 3, 3, 2, 2, 2, 3, 3, 3, 3, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1

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) = A368750(n,k) + A368751(n,k).

Example

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

Mathematica

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

Crossrefs

Cf. A000984 (row lengths), A068551 (row sums), A362030 and A368804 (binary words).

Cf. A368750 (atoms), A368751 (co-atoms), A368753 (defects).

Sequence in context: A352629 A106825 A156608 * A323827 A275850 A112505

Adjacent sequences: A368749 A368750 A368751 * A368753 A368754 A368755

Keyword

nonn,tabf

Author

Paolo Xausa, Jan 05 2024

Status

approved