A116469 Square array read by antidiagonals: T(m,n) = number of spanning trees in an m X n grid.

1, 1, 1, 1, 4, 1, 1, 15, 15, 1, 1, 56, 192, 56, 1, 1, 209, 2415, 2415, 209, 1, 1, 780, 30305, 100352, 30305, 780, 1, 1, 2911, 380160, 4140081, 4140081, 380160, 2911, 1, 1, 10864, 4768673, 170537640, 557568000, 170537640, 4768673, 10864, 1

Offset: 1

Author

Calculated by Hugo van der Sanden after a suggestion from Leroy Quet, Mar 20 2006

Keywords

• nonn
• tabl

Comments

This is the number of ways the points in an m X n grid can be connected to their orthogonal neighbors such that for any pair of points there is precisely one path connecting them.

a(n,n) = A007341(n).

a(m,n) = number of perfect mazes made from a grid of m X n cells. - Leroy Quet, Sep 08 2007

Also number of domino tilings of the (2m-1) X (2n-1) rectangle with upper left corner removed. For m=2, n=3 the 15 domino tilings of the 3 X 5 rectangle with upper left corner removed are:

. ._.___. . ._.___. . ._.___. . ._.___. . ._.___.

.|__|___| .|__|___| .| | |__| .|__|___| .| |__| |

| |_|___| | | | |_| | |||___| |_| |_| | ||__|_|

||__|___| |||_|_| ||__|___| |_|_|_| ||__|___|

. ._.___. . ._.___. . ._.___. . ._.___. . ._.___.

.|__|___| .|__|___| .| | |__| .|__|___| .|__|___|

| |_| | | | | | | | | | ||| | | |_| | | | | | |_| |

||__|_|| ||_|||_| ||__|_|| |__|_||| |||___|_|

. ._.___. . ._.___. . ._.___. . ._.___. . ._.___.

.|__| | | .|__| | | .| | | | | .|___| | | .|__|___|

| |_|_|| | | | ||_| | |||_|| |__| ||| |_|___| |

||__|___| |||_|_| ||__|___| |_|_|_| |_|___|_|

- Alois P. Heinz, Apr 15 2011

Each row (and column) of the square array is a divisibility sequence, i.e., if n divides m then a(n) divides a(m). It follows that the main diagonal, A007341, is also a divisibility sequence. Row k satisfies a linear recurrence of order 2^k. - Peter Bala, Apr 29 2014

Examples

			a(2,2) = 4, since we must have exactly 3 of the 4 possible connections: if we have all 4 there are multiple paths between points; if we have fewer some points will be isolated from others.
Array begins:
  1,   1,      1,         1,           1,              1, ...
  1,   4,     15,        56,         209,            780, ...
  1,  15,    192,      2415,       30305,         380160, ...
  1,  56,   2415,    100352,     4140081,      170537640, ...
  1, 209,  30305,   4140081,   557568000,    74795194705, ...
  1, 780, 380160, 170537640, 74795194705, 32565539635200, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 1..50, flattened
• Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. - From _N. J. A. Sloane_, May 27 2012

Crossrefs

Diagonal gives A007341. Rows and columns 1..10 give A000012, A001353, A006238, A003696, A003779, A139400, A334002, A334003, A334004, A334005.

Programs

Maple

Digits:=200;
T:=(m,n)->round(Re(evalf(simplify(expand(
mul(mul( 4*sin(h*Pi/(2*m))^2+4*sin(k*Pi/(2*n))^2, h=1..m-1), k=1..n-1)))))); # crude Maple program from N. J. A. Sloane, May 27 2012

Mathematica

T[m_, n_] := Product[4 Sin[h Pi/(2 m)]^2 + 4 Sin[k Pi/(2 n)]^2, {h, m - 1}, {k, n - 1}]; Flatten[Table[FullSimplify[T[k, r - k]], {r, 2, 10}, {k, 1, r - 1}]] (* Ben Branman, Mar 10 2013 *)

PARI

T(n,m) = polresultant(polchebyshev(n-1, 2, x/2), polchebyshev(m-1, 2, (4-x)/2)); \\ Michel Marcus, Apr 13 2020

Python

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A116469(n, k):
    if n == 1 or k == 1: return 1
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A116469(j + 1, i - j + 1) for i in range(9) for j in range(i + 1)])  # Seiichi Manyama, Apr 12 2020

Formula

T(m,n) = Product_{k=1..n-1} Product_{h=1..m-1} (4*sin(h*Pi/(2*m))^2 + 4*sin(k*Pi/(2*n))^2); [Kreweras] - N. J. A. Sloane, May 27 2012

Equivalently, T(n,m) = resultant( U(n-1,x/2), U(m-1,(4-x)/2) ) = Product_{k = 1..n-1} Product_{h = 1..m-1} (4 - 2*cos(h*Pi/m) - 2*cos(k*Pi/n)), where U(n,x) denotes the Chebyshev polynomial of the second kind. The divisibility properties of the array mentioned in the Comments follow from this representation. - Peter Bala, Apr 29 2014