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A370220 Irregular triangle T(n,k) read by rows: row n lists the positions of left parentheses for the properly nested string of parentheses encoded by A063171(n).

%I A370220 #29 Mar 27 2024 15:45:14
%S A370220 1,1,3,1,2,1,3,5,1,3,4,1,2,5,1,2,4,1,2,3,1,3,5,7,1,3,5,6,1,3,4,7,1,3,
%T A370220 4,6,1,3,4,5,1,2,5,7,1,2,5,6,1,2,4,7,1,2,4,6,1,2,4,5,1,2,3,7,1,2,3,6,
%U A370220 1,2,3,5,1,2,3,4,1,3,5,7,9,1,3,5,7,8,1,3,5,6,9
%N A370220 Irregular triangle T(n,k) read by rows: row n lists the positions of left parentheses for the properly nested string of parentheses encoded by A063171(n).
%C A370220 Knuth (2011) refers to these terms as z_k and notes that z_1, z_2, ..., z_m is one of the binomial(2*m,m) combinations of m >= 1 objects from the set {1, 2, ..., 2*m}, subject to the constraint that z_(k-1) < z_k < 2*k for 1 <= k <= m and assuming that z_0 = 0.
%D A370220 Donald E. Knuth, The Art of Computer Programming, Vol. 4A: Combinatorial Algorithms, Part 1, Addison-Wesley, 2011, Section 7.2.1.6, pp. 440-444. See also exercise 2, p. 471 and p. 781.
%H A370220 Paolo Xausa, <a href="/A370220/b370220.txt">Table of n, a(n) for n = 1..15521</a> (rows 1..2055 of the triangle, flattened).
%F A370220 T(n,k) = T(n,k+1) - A370219(n,k) - 1, for 1 <= k < A072643(n).
%e A370220 The following table lists z_k values for properly nested strings having lengths up to 8, along with d_k, p_k and c_k values from related combinatorial objects (see related sequences for more information). Cf. Knuth (2011), p. 442, Table 1.
%e A370220 .
%e A370220       | Properly |          | A370219 |         | A370221 | A370222
%e A370220       | Nested   | A063171  | d d d d | z z z z | p p p p | c c c c
%e A370220     n | String   |   (n)    | 1 2 3 4 | 1 2 3 4 | 1 2 3 4 | 1 2 3 4
%e A370220   ----+----------+----------+---------+---------+---------+---------
%e A370220     1 | ()       | 10       | 1       | 1       | 1       | 0
%e A370220     2 | ()()     | 1010     | 1 1     | 1 3     | 1 2     | 0 0
%e A370220     3 | (())     | 1100     | 0 2     | 1 2     | 2 1     | 0 1
%e A370220     4 | ()()()   | 101010   | 1 1 1   | 1 3 5   | 1 2 3   | 0 0 0
%e A370220     5 | ()(())   | 101100   | 1 0 2   | 1 3 4   | 1 3 2   | 0 0 1
%e A370220     6 | (())()   | 110010   | 0 2 1   | 1 2 5   | 2 1 3   | 0 1 0
%e A370220     7 | (()())   | 110100   | 0 1 2   | 1 2 4   | 2 3 1   | 0 1 1
%e A370220     8 | ((()))   | 111000   | 0 0 3   | 1 2 3   | 3 2 1   | 0 1 2
%e A370220     9 | ()()()() | 10101010 | 1 1 1 1 | 1 3 5 7 | 1 2 3 4 | 0 0 0 0
%e A370220    10 | ()()(()) | 10101100 | 1 1 0 2 | 1 3 5 6 | 1 2 4 3 | 0 0 0 1
%e A370220    11 | ()(())() | 10110010 | 1 0 2 1 | 1 3 4 7 | 1 3 2 4 | 0 0 1 0
%e A370220    12 | ()(()()) | 10110100 | 1 0 1 2 | 1 3 4 6 | 1 3 4 2 | 0 0 1 1
%e A370220    13 | ()((())) | 10111000 | 1 0 0 3 | 1 3 4 5 | 1 4 3 2 | 0 0 1 2
%e A370220    14 | (())()() | 11001010 | 0 2 1 1 | 1 2 5 7 | 2 1 3 4 | 0 1 0 0
%e A370220    15 | (())(()) | 11001100 | 0 2 0 2 | 1 2 5 6 | 2 1 4 3 | 0 1 0 1
%e A370220    16 | (()())() | 11010010 | 0 1 2 1 | 1 2 4 7 | 2 3 1 4 | 0 1 1 0
%e A370220    17 | (()()()) | 11010100 | 0 1 1 2 | 1 2 4 6 | 2 3 4 1 | 0 1 1 1
%e A370220    18 | (()(())) | 11011000 | 0 1 0 3 | 1 2 4 5 | 2 4 3 1 | 0 1 1 2
%e A370220    19 | ((()))() | 11100010 | 0 0 3 1 | 1 2 3 7 | 3 2 1 4 | 0 1 2 0
%e A370220    20 | ((())()) | 11100100 | 0 0 2 2 | 1 2 3 6 | 3 2 4 1 | 0 1 2 1
%e A370220    21 | ((()())) | 11101000 | 0 0 1 3 | 1 2 3 5 | 3 4 2 1 | 0 1 2 2
%e A370220    22 | (((()))) | 11110000 | 0 0 0 4 | 1 2 3 4 | 4 3 2 1 | 0 1 2 3
%t A370220 zlist[m_] := With[{r = 2*Range[2, m]}, Reverse[Map[Join[{1}, #] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] > 0 &]]]];
%t A370220 Array[Delete[zlist[#], 0] &, 5]
%t A370220 (* 2nd program: uses Algorithm Z from Knuth's TAOCP section 7.2.1.6, exercise 2 *)
%t A370220 zlist[m_] := Block[{z = 2*Range[m] - 1, j},
%t A370220     Reap[
%t A370220     While[True,
%t A370220         Sow[z];
%t A370220         If[z[[m-1]] < z[[m]] - 1,
%t A370220             z[[m]]--,
%t A370220             j = m - 1; z[[m]] = 2*m - 1;
%t A370220             While[j > 1 && z[[j-1]] == z[[j]] - 1, z[[j]] = 2*j - 1; j--];
%t A370220             If[j == 1,Break[]];
%t A370220             z[[j]]--]
%t A370220     ]][[2]][[1]]];
%t A370220 Join[{{1}}, Array[Delete[zlist[#], 0] &, 4, 2]]
%Y A370220 Cf. A000108, A063171, A072643 (row lengths).
%Y A370220 Cf. A370219, A370221, A370222, A370290 (row sums), A371409 (right parentheses).
%K A370220 nonn,tabf
%O A370220 1,3
%A A370220 _Paolo Xausa_, Feb 12 2024