# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a375354 Showing 1-1 of 1 %I A375354 #32 Oct 27 2024 12:12:20 %S A375354 1,1,2,1,4,20,1,8,104,1504,1,16,544,22208,948032,1,32,2848,329216, %T A375354 40930304,5204262912,1,64,14912,4883968,1772261888,666548862976, %U A375354 254112496082944,1,128,78080,72464384,76795762688,85575149027328,97392800416399360,111879597850371293184 %N A375354 T(m,n) is the number of suitably connected m X n Legendrian mosaics read by rows, with 1<=n<=m. %C A375354 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 A375354 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 A375354 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 A375354 Luc Ta, First 11 rows of the triangle, flattened %H A375354 Margaret Kipe, Samantha Pezzimenti, Leif Schaumann, Luc Ta, and Wing Hong Tony Wong, Bounds on the mosaic number of Legendrian knots, arXiv: 2410.08064 [math.GT], 2024. %H A375354 Seungsang Oh, Kyungpyo Hong, Ho Lee, and Hwa Jeong Lee, Quantum knots and the number of knot mosaics, arXiv: 1412.4460 [math.GT], 2014. %H A375354 S. Pezzimenti and A. Pandey, Geography of Legendrian knot mosaics, Journal of Knot Theory and its Ramifications, 31 (2022), article no. 2250002, 1-22. %H A375354 Index entries for sequences related to knots %F A375354 T(m,3) = A082761(m-1) for all m >= 1. - _Luc Ta_, Aug 20 2024 %e A375354 Triangle begins: %e A375354 1; %e A375354 1, 2; %e A375354 1, 4, 20; %e A375354 1, 8, 104, 1504; %e A375354 1, 16, 544, 22208, 948032; %e A375354 1, 32, 2848, 329216, 40930304, 5204262912; %e A375354 ... %e A375354 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 A375354 For all n >= 1, we have T(n,1) = 1 since the only suitably connected Legendrian mosaic with one column is empty. %t A375354 x[0] = o[0] = {{1}}; %t A375354 x[n_] := ArrayFlatten[{{x[n - 1], o[n - 1]}, {o[n - 1], x[n - 1]}}]; %t A375354 o[n_] := ArrayFlatten[{{o[n - 1], x[n - 1]}, {x[n - 1], 3*o[n - 1]}}]; %t A375354 legendrian[m_, n_] := If[m > 1 && n > 1, 2*Total[MatrixPower[x[m - 2] + o[m - 2], n - 2], 2], 1]; %t A375354 Flatten[ParallelTable[legendrian[m, n], {m, 1, 11}, {n, 1, m}]] (* _Luc Ta_, Aug 13 2024 *) %Y A375354 The main diagonal T(n,n) is A374947. %Y A375354 Cf. A000079, A082761, A374946, A375353, A261400. %K A375354 nonn,tabl %O A375354 1,3 %A A375354 _Margaret Kipe_, _Samantha Pezzimenti_, _Leif Schaumann_, _Luc Ta_, _Wing Hong Tony Wong_, Aug 13 2024 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE