%I #32 Oct 27 2024 12:12:20
%S 1,1,2,1,4,20,1,8,104,1504,1,16,544,22208,948032,1,32,2848,329216,
%T 40930304,5204262912,1,64,14912,4883968,1772261888,666548862976,
%U 254112496082944,1,128,78080,72464384,76795762688,85575149027328,97392800416399360,111879597850371293184
%N T(m,n) is the number of suitably connected m X n Legendrian mosaics read by rows, with 1<=n<=m.
%C An m X n Legendrian mosaic is an m X n array of the 10 tiles given in Figure 5 of Pezzimenti and Pandey. These tiles represent part of a Legendrian curve in the front projection. The condition of being suitably connected means that the connection points of each tile coincide with those of the contiguous tiles.
%C The Mathematica program below is based on the algorithm given in Theorem 1 of Oh, Hong, Lee, and Lee, adapted to the Legendrian setting: since Legendrian mosaic tiles omit the crossing tile T_9 used in general knot mosaics, the bottom-right submatrix of O_(k+1) is 3*O_k rather than 4*O_k. See Theorem 6 of Kipe et al.
%C T(m,2) = A375353(m,2) = A000079(m-1) for all m >= 2 since neither classical nor Legendrian link mosaics with only 2 columns or rows can use T_9 tiles.
%H Luc Ta, <a href="/A375354/b375354.txt">First 11 rows of the triangle, flattened</a>
%H Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, <a href="http://arxiv.org/abs/2410.08064">Bounds on the mosaic number of Legendrian knots</a>, arXiv: 2410.08064 [math.GT], 2024.
%H Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, <a href="http://arxiv.org/abs/1412.4460">Quantum knots and the number of knot mosaics</a>, arXiv: 1412.4460 [math.GT], 2014.
%H S. Pezzimenti and A. Pandey, <a href="https://doi.org/10.1142/S021821652250002X">Geography of Legendrian knot mosaics</a>, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.
%H <a href="/index/K#knots">Index entries for sequences related to knots</a>
%F T(m,3) = A082761(m-1) for all m >= 1. - _Luc Ta_, Aug 20 2024
%e Triangle begins:
%e 1;
%e 1, 2;
%e 1, 4, 20;
%e 1, 8, 104, 1504;
%e 1, 16, 544, 22208, 948032;
%e 1, 32, 2848, 329216, 40930304, 5204262912;
%e ...
%e T(2,2) = 2 since the only suitably connected 2 X 2 Legendrian mosaics are the empty mosaic and the mosaic depicting the Legendrian unknot with maximal Thurston-Bennequin invariant.
%e For all n >= 1, we have T(n,1) = 1 since the only suitably connected Legendrian mosaic with one column is empty.
%t x[0] = o[0] = {{1}};
%t x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
%t o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 3*o[n - 1]}}];
%t legendrian[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1];
%t Flatten[ParallelTable[legendrian[m, n], {m, 1, 11}, {n, 1, m}]] (* _Luc Ta_, Aug 13 2024 *)
%Y The main diagonal T(n,n) is A374947.
%Y Cf. A000079, A082761, A374946, A375353, A261400.
%K nonn,tabl
%O 1,3
%A _Margaret Kipe_, _Samantha Pezzimenti_, _Leif Schaumann_, _Luc Ta_, _Wing Hong Tony Wong_, Aug 13 2024