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A370222 Irregular triangle T(n,k) read by rows: row n gives the inversion table (see comments) of the permutation encoded by row n of A370221.

%I A370222 #37 Mar 27 2024 15:44:27
%S A370222 0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,1,1,0,1,2,0,0,0,0,0,0,0,1,0,0,1,0,0,0,
%T A370222 1,1,0,0,1,2,0,1,0,0,0,1,0,1,0,1,1,0,0,1,1,1,0,1,1,2,0,1,2,0,0,1,2,1,
%U A370222 0,1,2,2,0,1,2,3,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0
%N A370222 Irregular triangle T(n,k) read by rows: row n gives the inversion table (see comments) of the permutation encoded by row n of A370221.
%C A370222 Knuth (2011) uses the inversion table c_1, c_2, ..., c_k (defined so that exactly c_k elements to the right of k are less than k) to encode the permutation given by row n of A370221.
%C A370222 This way c_1 = 0 and 0 <= c_(k+1) <= c_k + 1 for 1 <= k < m, where m >= 1 is half the length of the corresponding properly nested string of parentheses (see example).
%C A370222 The concatenation of terms in each row from row n = 23714 to 82498 (corresponding to strings of length 22, excluding the last one) gives the A000108(11) - 1 = 58785 terms of the finite sequence A239903.
%D A370222 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 A370222 Paolo Xausa, <a href="/A370222/b370222.txt">Table of n, a(n) for n = 1..15521</a> (rows 1..2055 of the triangle, flattened).
%F A370222 T(n,k) = 2*k - 1 - A370220(n,k).
%e A370222 The following table lists c_k values for properly nested strings having lengths up to 8, along with d_k, z_k and p_k values from related combinatorial objects (see related sequences for more information). Cf. Knuth (2011), p. 442, Table 1.
%e A370222 .
%e A370222       | Properly |          | A370219 | A370220 | A370221 |
%e A370222       | Nested   | A063171  | d d d d | z z z z | p p p p | c c c c
%e A370222     n | String   |   (n)    | 1 2 3 4 | 1 2 3 4 | 1 2 3 4 | 1 2 3 4
%e A370222   ----+----------+----------+---------+---------+---------+---------
%e A370222     1 | ()       | 10       | 1       | 1       | 1       | 0
%e A370222     2 | ()()     | 1010     | 1 1     | 1 3     | 1 2     | 0 0
%e A370222     3 | (())     | 1100     | 0 2     | 1 2     | 2 1     | 0 1
%e A370222     4 | ()()()   | 101010   | 1 1 1   | 1 3 5   | 1 2 3   | 0 0 0
%e A370222     5 | ()(())   | 101100   | 1 0 2   | 1 3 4   | 1 3 2   | 0 0 1
%e A370222     6 | (())()   | 110010   | 0 2 1   | 1 2 5   | 2 1 3   | 0 1 0
%e A370222     7 | (()())   | 110100   | 0 1 2   | 1 2 4   | 2 3 1   | 0 1 1
%e A370222     8 | ((()))   | 111000   | 0 0 3   | 1 2 3   | 3 2 1   | 0 1 2
%e A370222     9 | ()()()() | 10101010 | 1 1 1 1 | 1 3 5 7 | 1 2 3 4 | 0 0 0 0
%e A370222    10 | ()()(()) | 10101100 | 1 1 0 2 | 1 3 5 6 | 1 2 4 3 | 0 0 0 1
%e A370222    11 | ()(())() | 10110010 | 1 0 2 1 | 1 3 4 7 | 1 3 2 4 | 0 0 1 0
%e A370222    12 | ()(()()) | 10110100 | 1 0 1 2 | 1 3 4 6 | 1 3 4 2 | 0 0 1 1
%e A370222    13 | ()((())) | 10111000 | 1 0 0 3 | 1 3 4 5 | 1 4 3 2 | 0 0 1 2
%e A370222    14 | (())()() | 11001010 | 0 2 1 1 | 1 2 5 7 | 2 1 3 4 | 0 1 0 0
%e A370222    15 | (())(()) | 11001100 | 0 2 0 2 | 1 2 5 6 | 2 1 4 3 | 0 1 0 1
%e A370222    16 | (()())() | 11010010 | 0 1 2 1 | 1 2 4 7 | 2 3 1 4 | 0 1 1 0
%e A370222    17 | (()()()) | 11010100 | 0 1 1 2 | 1 2 4 6 | 2 3 4 1 | 0 1 1 1
%e A370222    18 | (()(())) | 11011000 | 0 1 0 3 | 1 2 4 5 | 2 4 3 1 | 0 1 1 2
%e A370222    19 | ((()))() | 11100010 | 0 0 3 1 | 1 2 3 7 | 3 2 1 4 | 0 1 2 0
%e A370222    20 | ((())()) | 11100100 | 0 0 2 2 | 1 2 3 6 | 3 2 4 1 | 0 1 2 1
%e A370222    21 | ((()())) | 11101000 | 0 0 1 3 | 1 2 3 5 | 3 4 2 1 | 0 1 2 2
%e A370222    22 | (((()))) | 11110000 | 0 0 0 4 | 1 2 3 4 | 4 3 2 1 | 0 1 2 3
%t A370222 clist[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Join[{0}, r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
%t A370222 Array[Delete[clist[#], 0] &, 5]
%t A370222 (* 2nd program: uses Algorithm Z from Knuth's TAOCP section 7.2.1.6, exercise 2 *)
%t A370222 zlist[m_] := Block[{z = 2*Range[m] - 1, j},
%t A370222     Reap[
%t A370222     While[True,
%t A370222         Sow[z];
%t A370222         If[z[[m-1]] < z[[m]] - 1,
%t A370222             z[[m]]--,
%t A370222             j = m - 1; z[[m]] = 2*m - 1;
%t A370222             While[j > 1 && z[[j-1]] == z[[j]] - 1, z[[j]] = 2*j - 1; j--];
%t A370222             If[j == 1,Break[]];
%t A370222             z[[j]]--]
%t A370222     ]][[2]][[1]]];
%t A370222 Join[{{0}}, Table[Delete[Map[2*Range[n] - 1 - # &, zlist[n]], 0], {n, 2, 5}]] (* _Paolo Xausa_, Mar 25 2024 *)
%Y A370222 Cf. A000108, A063171, A072643 (row lengths), A239903.
%Y A370222 Cf. A370219, A370220, A370221, A370292 (row sums).
%K A370222 nonn,tabf
%O A370222 1,20
%A A370222 _Paolo Xausa_, Feb 12 2024