A375354 T(m,n) is the number of suitably connected m X n Legendrian mosaics read by rows, with 1<=n<=m.

1, 1, 2, 1, 4, 20, 1, 8, 104, 1504, 1, 16, 544, 22208, 948032, 1, 32, 2848, 329216, 40930304, 5204262912, 1, 64, 14912, 4883968, 1772261888, 666548862976, 254112496082944, 1, 128, 78080, 72464384, 76795762688, 85575149027328, 97392800416399360, 111879597850371293184

Offset

1,3

Comments

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.

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.

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.

Links

Luc Ta, First 11 rows of the triangle, flattened

Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014.

S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22.

Index entries for sequences related to knots

Formula

T(m,3) = A082761(m-1) for all m >= 1. - Luc Ta, Aug 20 2024

Example

Triangle begins:
  1;
  1,  2;
  1,  4,   20;
  1,  8,  104,   1504;
  1, 16,  544,  22208,   948032;
  1, 32, 2848, 329216, 40930304, 5204262912;
  ...
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.
For all n >= 1, we have T(n, 1) = 1 since the only suitably connected Legendrian mosaic with one column is empty.

Mathematica

x[0] = o[0] = {{1}};
x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}];
o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 3*o[n - 1]}}];
legendrian[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1];
Flatten[ParallelTable[legendrian[m, n], {m, 1, 11}, {n, 1, m}]] (* Luc Ta, Aug 13 2024 *)

Crossrefs

The diagonal subsequence T(n,n) is A374947.

Cf. A000079, A082761, A374946, A375353, A261400.

Sequence in context: A032105 A259472 A354055 * A053565 A116603 A354056

Adjacent sequences: A375351 A375352 A375353 * A375355 A375356 A375357

Keyword

nonn,tabl,new

Author

Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, Wing Hong Tony Wong, Aug 13 2024

Status

approved