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A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).

%I A321172 #98 Jun 29 2025 17:46:31
%S A321172 1,1,0,1,2,6,1,0,14,0,1,4,37,154,1072,1,0,92,0,5320,0,1,8,236,1696,
%T A321172 32675,301384,4638576,1,0,596,0,175294,0,49483138,0,1,16,1517,18684,
%U A321172 1024028,17066492,681728204,13916993782,467260456608
%N A321172 Triangle read by rows: T(m,n) = number of Hamiltonian cycles on m X n grid of points (m >= 2, 2 <= n <= m).
%C A321172 Orientation of the path is not important; you can start going either clockwise or counterclockwise. Paths related by symmetries are considered distinct.
%C A321172 The m X n grid of points when drawn forms a (m-1) X (n-1) rectangle of cells, so m-1 and n-1 are sometimes used as indices instead of m and n (see, e. g., Kreweras' reference below).
%C A321172 These cycles are also called "closed non-intersecting rook's tours" on m X n chess board.
%H A321172 Huaide Cheng, <a href="/A321172/b321172.txt">Rows n=2..17 of triangle, flattened</a>
%H A321172 Robert Ferréol, <a href="/A321172/a321172.gif">The T(4,5)=14 hamiltonian cycles on 4 X 5 square grid of points</a>; the T(5,6) = 154 cycles on 5 X 6 grid <a href="/A321172/a321172_1.gif">are reduced to 44 different forms</a>.
%H A321172 Germain Kreweras, <a href="https://doi.org/10.1016/0195-6698(92)90005-K">Dénombrement des cycles hamiltoniens dans un rectangle quadrillé</a>, European Journal of Combinatorics, Volume 13, Issue 6 (1992), page 476.
%H A321172 Robert Stoyan and Volker Strehl, <a href="https://www.emis.de/journals/SLC/wpapers/s34erlangen.html">Enumeration of Hamiltonian Circuits in Rectangular Grids</a>, Séminaire Lotharingien de Combinatoire, B34f (1995), 21 pp.
%H A321172 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%H A321172 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HamiltonianCycle.html">Hamiltonian Cycle</a>.
%F A321172 T(m,n) = T(n,m).
%F A321172 T(2m+1,2n+1) = 0.
%F A321172 T(2n,2n) = A003763(n).
%F A321172 T(n,n+1) = T(n+1,n) = A222200(n).
%F A321172 G. functions , G_m(x)=Sum {n>=0} T(m,n-2)*x^n after Stoyan's link p. 18 :
%F A321172 G_2(x) = 1/(1-x) = 1+x+x^2+...
%F A321172 G_3(x) = 1/(1-2*x^2) = 1+2*x^2+4*x^4+...
%F A321172 G_4(x) = 1/(1-2*x-2*x^2+2*x^3-x^4) = 1+2*x+6*x^2+...
%F A321172 G_5(x) = (1+3*x^2)/(1-11*x^2-2*x^6) = 1+14*x^2+154*x^4+...
%e A321172 T(5,4)=14 is illustrated in the links above.
%e A321172 Table starts:
%e A321172 =================================================================
%e A321172 m\n|  2    3      4       5         6           7            8
%e A321172 ---|-------------------------------------------------------------
%e A321172 2  |  1    1      1       1         1           1            1
%e A321172 3  |  1    0      2       0         4           0            8
%e A321172 4  |  1    2      6      14        37          92          236
%e A321172 5  |  1    0     14       0       154           0         1696
%e A321172 6  |  1    4     37     154      1072        5320        32675
%e A321172 7  |  1    0     92       0      5320           0       301384
%e A321172 8  |  1    8    236    1696     32675      301384      4638576
%e A321172 The table is symmetric, so it is completely described by its lower-left half.
%o A321172 (Python)
%o A321172 # Program due to Laurent Jouhet-Reverdy
%o A321172 def cycle(m, n):
%o A321172      if (m%2==1 and n%2==1): return 0
%o A321172      grid = [[0]*n for _ in range(m)]
%o A321172      grid[0][0] = 1; grid[1][0] = 1
%o A321172      counter = [0]; stop = m*n-1
%o A321172      def run(i, j, nb_points):
%o A321172          for ni, nj in [(i-1, j), (i+1, j), (i, j+1), (i, j-1)] :
%o A321172              if  0<=ni<=m-1 and 0<=nj<=n-1 and grid[ni][nj]==0 and (ni,nj)!=(0,1):
%o A321172                  grid[ni][nj] = 1
%o A321172                  run(ni, nj, nb_points+1)
%o A321172                  grid[ni][nj] = 0
%o A321172              elif (ni,nj)==(0,1) and nb_points==stop:
%o A321172                  counter[0] += 1
%o A321172      run(1, 0, 2)
%o A321172      return counter[0]
%o A321172 L=[];n=7#maximum for a time < 1 mn
%o A321172 for i in range(2,n):
%o A321172     for j in range(2,i+1):
%o A321172        L.append(cycle(i,j))
%o A321172 print(L)
%Y A321172 Row/column k=4..12 are: (with interspersed zeros for odd k): A006864, A006865, A145401, A145416, A145418, A160149, A180504, A180505, A213813.
%Y A321172 Cf. A003763 (bisection of main diagonal), A222200 (subdiagonal), A231829, A270273, A332307.
%Y A321172 T(n,2n) gives A333864.
%K A321172 nonn,tabl
%O A321172 2,5
%A A321172 _Robert FERREOL_, Jan 10 2019
%E A321172 More terms from _Pontus von Brömssen_, Feb 15 2021