OFFSET
1,3
COMMENTS
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.
REFERENCES
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.
LINKS
Paolo Xausa, Table of n, a(n) for n = 1..15521 (rows 1..2055 of the triangle, flattened).
EXAMPLE
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.
.
| Nested | A063171 | d d d d | z z z z | p p p p | c c c c
n | String | (n) | 1 2 3 4 | 1 2 3 4 | 1 2 3 4 | 1 2 3 4
----+----------+----------+---------+---------+---------+---------
1 | () | 10 | 1 | 1 | 1 | 0
2 | ()() | 1010 | 1 1 | 1 3 | 1 2 | 0 0
3 | (()) | 1100 | 0 2 | 1 2 | 2 1 | 0 1
4 | ()()() | 101010 | 1 1 1 | 1 3 5 | 1 2 3 | 0 0 0
5 | ()(()) | 101100 | 1 0 2 | 1 3 4 | 1 3 2 | 0 0 1
6 | (())() | 110010 | 0 2 1 | 1 2 5 | 2 1 3 | 0 1 0
7 | (()()) | 110100 | 0 1 2 | 1 2 4 | 2 3 1 | 0 1 1
8 | ((())) | 111000 | 0 0 3 | 1 2 3 | 3 2 1 | 0 1 2
9 | ()()()() | 10101010 | 1 1 1 1 | 1 3 5 7 | 1 2 3 4 | 0 0 0 0
10 | ()()(()) | 10101100 | 1 1 0 2 | 1 3 5 6 | 1 2 4 3 | 0 0 0 1
11 | ()(())() | 10110010 | 1 0 2 1 | 1 3 4 7 | 1 3 2 4 | 0 0 1 0
12 | ()(()()) | 10110100 | 1 0 1 2 | 1 3 4 6 | 1 3 4 2 | 0 0 1 1
13 | ()((())) | 10111000 | 1 0 0 3 | 1 3 4 5 | 1 4 3 2 | 0 0 1 2
14 | (())()() | 11001010 | 0 2 1 1 | 1 2 5 7 | 2 1 3 4 | 0 1 0 0
15 | (())(()) | 11001100 | 0 2 0 2 | 1 2 5 6 | 2 1 4 3 | 0 1 0 1
16 | (()())() | 11010010 | 0 1 2 1 | 1 2 4 7 | 2 3 1 4 | 0 1 1 0
17 | (()()()) | 11010100 | 0 1 1 2 | 1 2 4 6 | 2 3 4 1 | 0 1 1 1
18 | (()(())) | 11011000 | 0 1 0 3 | 1 2 4 5 | 2 4 3 1 | 0 1 1 2
19 | ((()))() | 11100010 | 0 0 3 1 | 1 2 3 7 | 3 2 1 4 | 0 1 2 0
20 | ((())()) | 11100100 | 0 0 2 2 | 1 2 3 6 | 3 2 4 1 | 0 1 2 1
21 | ((()())) | 11101000 | 0 0 1 3 | 1 2 3 5 | 3 4 2 1 | 0 1 2 2
22 | (((()))) | 11110000 | 0 0 0 4 | 1 2 3 4 | 4 3 2 1 | 0 1 2 3
MATHEMATICA
zlist[m_] := With[{r = 2*Range[2, m]}, Reverse[Map[Join[{1}, #] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] > 0 &]]]];
Array[Delete[zlist[#], 0] &, 5]
(* 2nd program: uses Algorithm Z from Knuth's TAOCP section 7.2.1.6, exercise 2 *)
zlist[m_] := Block[{z = 2*Range[m] - 1, j},
Reap[
While[True,
Sow[z];
If[z[[m-1]] < z[[m]] - 1,
z[[m]]--,
j = m - 1; z[[m]] = 2*m - 1;
While[j > 1 && z[[j-1]] == z[[j]] - 1, z[[j]] = 2*j - 1; j--];
If[j == 1, Break[]];
z[[j]]--]
]][[2]][[1]]];
Join[{{1}}, Array[Delete[zlist[#], 0] &, 4, 2]]
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Paolo Xausa, Feb 12 2024
STATUS
approved