This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370221 #21 Feb 18 2024 17:20:49
%S A370221 1,1,2,2,1,1,2,3,1,3,2,2,1,3,2,3,1,3,2,1,1,2,3,4,1,2,4,3,1,3,2,4,1,3,
%T A370221 4,2,1,4,3,2,2,1,3,4,2,1,4,3,2,3,1,4,2,3,4,1,2,4,3,1,3,2,1,4,3,2,4,1,
%U A370221 3,4,2,1,4,3,2,1,1,2,3,4,5,1,2,3,5,4,1,2,4,3,5
%N A370221 Irregular triangle T(n,k) read by rows: row n lists the values encoding a permutation (see comments) related to the properly nested string of parentheses encoded by A063171(n).
%C A370221 Knuth (2011) refers to these terms as p_k and defines them so that, in a properly nested string of parentheses, the k-th right parenthesis matches the p_k-th left parenthesis.
%C A370221 A370222 gives the corresponding values of a related inversion table.
%D A370221 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.
%H A370221 Paolo Xausa, <a href="/A370221/b370221.txt">Table of n, a(n) for n = 1..15521</a> (rows 1..2055 of the triangle, flattened).
%e A370221 The following table lists p_k values for properly nested strings having lengths up to 8, along with d_k, z_k and c_k values from related combinatorial objects (see related sequences for more information). Cf. Knuth (2011), p. 442, Table 1.
%e A370221 .
%e A370221 | Properly | | A370219 | A370220 | | A370222
%e A370221 | Nested | A063171 | d d d d | z z z z | p p p p | c c c c
%e A370221 n | String | (n) | 1 2 3 4 | 1 2 3 4 | 1 2 3 4 | 1 2 3 4
%e A370221 ----+----------+----------+---------+---------+---------+---------
%e A370221 1 | () | 10 | 1 | 1 | 1 | 0
%e A370221 2 | ()() | 1010 | 1 1 | 1 3 | 1 2 | 0 0
%e A370221 3 | (()) | 1100 | 0 2 | 1 2 | 2 1 | 0 1
%e A370221 4 | ()()() | 101010 | 1 1 1 | 1 3 5 | 1 2 3 | 0 0 0
%e A370221 5 | ()(()) | 101100 | 1 0 2 | 1 3 4 | 1 3 2 | 0 0 1
%e A370221 6 | (())() | 110010 | 0 2 1 | 1 2 5 | 2 1 3 | 0 1 0
%e A370221 7 | (()()) | 110100 | 0 1 2 | 1 2 4 | 2 3 1 | 0 1 1
%e A370221 8 | ((())) | 111000 | 0 0 3 | 1 2 3 | 3 2 1 | 0 1 2
%e A370221 9 | ()()()() | 10101010 | 1 1 1 1 | 1 3 5 7 | 1 2 3 4 | 0 0 0 0
%e A370221 10 | ()()(()) | 10101100 | 1 1 0 2 | 1 3 5 6 | 1 2 4 3 | 0 0 0 1
%e A370221 11 | ()(())() | 10110010 | 1 0 2 1 | 1 3 4 7 | 1 3 2 4 | 0 0 1 0
%e A370221 12 | ()(()()) | 10110100 | 1 0 1 2 | 1 3 4 6 | 1 3 4 2 | 0 0 1 1
%e A370221 13 | ()((())) | 10111000 | 1 0 0 3 | 1 3 4 5 | 1 4 3 2 | 0 0 1 2
%e A370221 14 | (())()() | 11001010 | 0 2 1 1 | 1 2 5 7 | 2 1 3 4 | 0 1 0 0
%e A370221 15 | (())(()) | 11001100 | 0 2 0 2 | 1 2 5 6 | 2 1 4 3 | 0 1 0 1
%e A370221 16 | (()())() | 11010010 | 0 1 2 1 | 1 2 4 7 | 2 3 1 4 | 0 1 1 0
%e A370221 17 | (()()()) | 11010100 | 0 1 1 2 | 1 2 4 6 | 2 3 4 1 | 0 1 1 1
%e A370221 18 | (()(())) | 11011000 | 0 1 0 3 | 1 2 4 5 | 2 4 3 1 | 0 1 1 2
%e A370221 19 | ((()))() | 11100010 | 0 0 3 1 | 1 2 3 7 | 3 2 1 4 | 0 1 2 0
%e A370221 20 | ((())()) | 11100100 | 0 0 2 2 | 1 2 3 6 | 3 2 4 1 | 0 1 2 1
%e A370221 21 | ((()())) | 11101000 | 0 0 1 3 | 1 2 3 5 | 3 4 2 1 | 0 1 2 2
%e A370221 22 | (((()))) | 11110000 | 0 0 0 4 | 1 2 3 4 | 4 3 2 1 | 0 1 2 3
%t A370221 slist[m_] := Reverse[Select[Permutations[PadLeft[Table[-1, m], 2*m, 1]], Min[Accumulate[#]] >= 0 &]];
%t A370221 plist[s_] := Flatten[Reap[Module[{p, p0 = Flatten[Position[s, -1]], p1 = Flatten[Position[s, 1]], p1r}, p1r = p1; For[i = 1, i <= Length[p0], i++, p = Max[Select[p1r, # < p0[[i]] &]]; Sow[Position[p1, p]]; p1r = DeleteCases[p1r, p]]]][[2,1]]];
%t A370221 Array[Delete[Map[plist, slist[#]], 0] &, 5]
%Y A370221 Cf. A000108, A063171, A072643 (row lengths).
%Y A370221 Cf. A370219, A370220, A370222, A370291 (row sums).
%K A370221 nonn,tabf
%O A370221 1,3
%A A370221 _Paolo Xausa_, Feb 12 2024