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Irregular triangle read by rows: T(n,k) is the number of 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.
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%I #33 Mar 06 2024 17:48:00

%S 0,1,0,0,1,1,2,1,0,0,0,1,0,0,1,1,2,1,1,1,2,2,3,2,1,2,1,1,0,0,0,0,1,0,

%T 0,0,1,0,0,1,1,2,1,0,0,0,1,0,0,1,1,2,1,1,1,2,2,3,2,1,2,1,1,1,1,1,2,1,

%U 1,2,2,3,2,2,2,3,3,4,3,2,3,2,2,1,1,2,2,3,2,1,2,1,1,1,2,1,1,1

%N Irregular triangle read by rows: T(n,k) is the number of 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.

%C A balanced string of parentheses is composed of the same number of left and right parentheses. E.g., "(())()" and "))(()(" are balanced, while "((()((()" is not.

%C A balanced string can be uniquely split into substrings s_1, s_2, ..., s_k, where s_i is either an atom (i.e., a balanced string, as short as possible, beginning with "(" and ending with ")"), or a co-atom (the reverse of an atom). Please note that, here, an atom is defined as a shortest possible properly nested substring: the string "()()" is composed of two contiguous atoms, while Knuth (2011) considers it a single atom.

%C To count for the number of atoms/co-atoms, we begin from the leftmost character of the string and proceed to the right, adding 1 to a counter when "(" is encountered, and subtracting 1 when ")" is encountered. Each time the counter reaches 0, we have found either an atom or a co-atom, depending on whether the starting character was a "(" or a ")", respectively.

%C For example, the string ")(()()(())))((" can be split into ")(" (co-atom), "()" (atom), "()" (atom), "(())" (atom) and "))((" (co-atom).

%C If ")" is encoded by 0 and "(" is encoded by 1, the triangle with the balanced strings is given by A368804 (and, converted to decimal, by A362030).

%D 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.

%H Paolo Xausa, <a href="/A368750/b368750.txt">Table of n, a(n) for n = 1..17576</a> (rows 1..8 of the triangle, flattened).

%F T(n,k) = A368752(n,k) - A368751(n,k).

%e Triangle begins:

%e [1] 0 1;

%e [2] 0 0 1 1 2 1;

%e [3] 0 0 0 1 0 0 1 1 2 1 1 1 2 2 3 2 1 2 1 1;

%e ...

%e The strings corresponding to row 2, in reverse lexicographical order, are:

%e "))((" (0 atoms),

%e ")()(" (0 atoms),

%e ")(()" (1 atom),

%e "())(" (1 atom),

%e "()()" (2 atoms) and

%e "(())" (1 atom).

%t strings[n_] := Permutations[PadLeft[Table[1, n], 2*n, -1]];

%t Array[Map[SequenceCount[Accumulate[#], {1, 0}] &, strings[#]] &, 5]

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

%Y Cf. A368751 (co-atoms), A368752 (all atoms), A368753 (defects).

%K nonn,tabf

%O 1,7

%A _Paolo Xausa_, Jan 05 2024