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

Author

Paolo Xausa, Jan 05 2024

Keywords

• nonn
• tabf

Comments

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

Examples

			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).
		

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).

Crossrefs

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

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

Programs

Mathematica

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

Formula

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