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

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, 32675, 301384, 4638576, 1, 0, 596, 0, 175294, 0, 49483138, 0, 1, 16, 1517, 18684, 1024028, 17066492, 681728204, 13916993782, 467260456608

Offset: 2

Author

Robert FERREOL, Jan 10 2019

Keywords

• nonn
• tabl

Comments

Orientation of the path is not important; you can start going either clockwise or counterclockwise. Paths related by symmetries are considered distinct.

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).

These cycles are also called "closed non-intersecting rook's tours" on m X n chess board.

Examples

			T(5,4)=14 is illustrated in the links above.
Table starts:
=================================================================
m\n|  2    3      4       5         6           7            8
---|-------------------------------------------------------------
2  |  1    1      1       1         1           1            1
3  |  1    0      2       0         4           0            8
4  |  1    2      6      14        37          92          236
5  |  1    0     14       0       154           0         1696
6  |  1    4     37     154      1072        5320        32675
7  |  1    0     92       0      5320           0       301384
8  |  1    8    236    1696     32675      301384      4638576
The table is symmetric, so it is completely described by its lower-left half.
		

Links

• Huaide Cheng, Rows n=2..17 of triangle, flattened
• Robert Ferréol, The T(4,5)=14 hamiltonian cycles on 4 X 5 square grid of points; the T(5,6) = 154 cycles on 5 X 6 grid are reduced to 44 different forms.
• Germain Kreweras, Dénombrement des cycles hamiltoniens dans un rectangle quadrillé, European Journal of Combinatorics, Volume 13, Issue 6 (1992), page 476.
• Robert Stoyan and Volker Strehl, Enumeration of Hamiltonian Circuits in Rectangular Grids, Séminaire Lotharingien de Combinatoire, B34f (1995), 21 pp.
• Eric Weisstein's World of Mathematics, Grid Graph.
• Eric Weisstein's World of Mathematics, Hamiltonian Cycle.

Crossrefs

Row/column k=4..12 are: (with interspersed zeros for odd k): A006864, A006865, A145401, A145416, A145418, A160149, A180504, A180505, A213813.

Cf. A003763 (bisection of main diagonal), A222200 (subdiagonal), A231829, A270273, A332307.

T(n,2n) gives A333864.

Programs

Python

# Program due to Laurent Jouhet-Reverdy
def cycle(m, n):
     if (m%2==1 and n%2==1): return 0
     grid = [[0]*n for _ in range(m)]
     grid[0][0] = 1; grid[1][0] = 1
     counter = [0]; stop = m*n-1
     def run(i, j, nb_points):
         for ni, nj in [(i-1, j), (i+1, j), (i, j+1), (i, j-1)] :
             if  0<=ni<=m-1 and 0<=nj<=n-1 and grid[ni][nj]==0 and (ni,nj)!=(0,1):
                 grid[ni][nj] = 1
                 run(ni, nj, nb_points+1)
                 grid[ni][nj] = 0
             elif (ni,nj)==(0,1) and nb_points==stop:
                 counter[0] += 1
     run(1, 0, 2)
     return counter[0]
L=[];n=7#maximum for a time < 1 mn
for i in range(2,n):
    for j in range(2,i+1):
       L.append(cycle(i,j))
print(L)

Formula

T(m,n) = T(n,m).

T(2m+1,2n+1) = 0.

T(2n,2n) = A003763(n).

T(n,n+1) = T(n+1,n) = A222200(n).

G. functions , G_m(x)=Sum {n>=0} T(m,n-2)*x^n after Stoyan's link p. 18 :

G_2(x) = 1/(1-x) = 1+x+x^2+...

G_3(x) = 1/(1-2*x^2) = 1+2*x^2+4*x^4+...

G_4(x) = 1/(1-2*x-2*x^2+2*x^3-x^4) = 1+2*x+6*x^2+...

G_5(x) = (1+3*x^2)/(1-11*x^2-2*x^6) = 1+14*x^2+154*x^4+...

Extensions

More terms from Pontus von Brömssen, Feb 15 2021