A340425 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of spanning trees of odd Aztec rectangle of order (n, k).

1, 4, 4, 16, 192, 16, 64, 8960, 8960, 64, 256, 417792, 4542720, 417792, 256, 1024, 19480576, 2280570880, 2280570880, 19480576, 1024, 4096, 908328960, 1143668117504, 12116689944576, 1143668117504, 908328960, 4096

Offset: 1

Author

Seiichi Manyama, Jan 07 2021

Keywords

• nonn
• tabl

Examples

			Square array begins:
    1,        4,            16,                64,                    256, ...
    4,      192,          8960,            417792,               19480576, ...
   16,     8960,       4542720,        2280570880,          1143668117504, ...
   64,   417792,    2280570880,    12116689944576,      64046643170770944, ...
  256, 19480576, 1143668117504, 64046643170770944, 3544863978266468352000, ...
		

References

• Mihai Ciucu, Matchings and applications. Department of Mathematics, Indiana University, Bloomington, IN 47405. See Lecture 12.

Crossrefs

Main diagonal gives A340352.

Cf. A340427.

Programs

PARI

default(realprecision, 120);
{T(n,k) = round(4^(2*n*k-n-k)*prod(a=1, n-1, prod(b=1, k-1, 1-(sin(a*Pi/(2*n))*sin(b*Pi/(2*k)))^2)))}

Python

# Using graphillion
from graphillion import GraphSet
def make_OD_nk(n, k):
    n += 1
    k += 1
    grids = []
    s = k * n
    for i in range(1, k * n, k):
        for j in range(1, k):
            a, b = i + j - 1, i + j
            c = s + a
            if i > 1:
                grids.extend([(c - k, a), (c - k, b)])
            if i < k * (n - 1) + 1:
                grids.extend([(c, a), (c, b)])
    return grids
def A340425(n, k):
    universe = make_OD_nk(n, k)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A340425(j + 1, i - j + 1) for i in range(7) for j in range(i + 1)])

Formula

T(n,k) = T(k,n).

T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * sin(b*Pi/(2*k))^2).

T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - cos(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).

T(n,k) = 4^(2*n*k-n-k) * Product_{a=1..n-1} Product_{b=1..k-1} (1 - sin(a*Pi/(2*n))^2 * cos(b*Pi/(2*k))^2).