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 A269565 #48 Feb 16 2025 08:33:30 %S A269565 1,2,2,6,8,6,24,60,60,24,120,816,1512,816,120,720,17520,83520,83520, %T A269565 17520,720,5040,550080,8869680,22394880,8869680,550080,5040,40320, %U A269565 23839200,1621680480,13346910720,13346910720,1621680480,23839200,40320 %N A269565 Array read by antidiagonals: T(n,m) is the number of (directed) Hamiltonian paths in K_n X K_m. %C A269565 Equivalently, the number of directed Hamiltonian paths on the n X m rook graph. %C A269565 Conjecture: T(n,m) mod n!*m! = 0. - _Mikhail Kurkov_, Feb 08 2019 %C A269565 The above conjecture is true since a path defines an ordering on the rows and columns by the order in which they are first visited by the path. Every permutation of rows and columns therefore gives a different path. - _Andrew Howroyd_, Feb 08 2021 %H A269565 Andrew Howroyd, <a href="/A269565/b269565.txt">Table of n, a(n) for n = 1..96</a> %H A269565 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianPath.html">Hamiltonian Path</a> %H A269565 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RookGraph.html">Rook Graph</a> %F A269565 From _Andrew Howroyd_, Oct 20 2019: (Start) %F A269565 T(n,m) = T(m,n). %F A269565 T(n,1) = n!. (End) %e A269565 Array begins: %e A269565 =========================================================== %e A269565 n\m| 1 2 3 4 5 %e A269565 ---+------------------------------------------------------- %e A269565 1 | 1, 2, 6, 24, 120, ... %e A269565 2 | 2, 8, 60, 816, 17520, ... %e A269565 3 | 6, 60, 1512, 83520, 8869680, ... %e A269565 4 | 24, 816, 83520, 22394880, 13346910720, ... %e A269565 5 | 120, 17520, 8869680, 13346910720, 50657369241600, ... %e A269565 ... %Y A269565 Main diagonal is A096970. %Y A269565 Columns 2..3 are A096121, A329319. %Y A269565 Cf. A286418, A269562. %K A269565 nonn,tabl %O A269565 1,2 %A A269565 _Andrew Howroyd_, Feb 29 2016