Search: a334005 -id:a334005
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A116469
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Square array read by antidiagonals: T(m,n) = number of spanning trees in an m X n grid.
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+10
25
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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
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OFFSET
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1,5
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COMMENTS
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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(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:
. .___.___. . .___.___. . .___.___. . .___.___. . .___.___.
._|___|___| ._|___|___| ._| | |___| ._|___|___| ._| |___| |
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|_|___|___| |_|_|_|___| |_|___|___| |___|_|___| |_|___|___|
. .___.___. . .___.___. . .___.___. . .___.___. . .___.___.
._|___|___| ._|___|___| ._| | |___| ._|___|___| ._|___|___|
| |___| | | | | | | | | | |_|_| | | |___| | | | | | |___| |
|_|___|_|_| |_|_|_|_|_| |_|___|_|_| |___|_|_|_| |_|_|___|_|
. .___.___. . .___.___. . .___.___. . .___.___. . .___.___.
._|___| | | ._|___| | | ._| | | | | ._|___| | | ._|___|___|
| |___|_|_| | | | |_|_| | |_|_|_|_| |___| |_|_| |___|___| |
|_|___|___| |_|_|_|___| |_|___|___| |___|_|___| |___|___|_|
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
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LINKS
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FORMULA
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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
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EXAMPLE
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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, ...
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Python)
# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
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()
(PARI) T(n, m) = polresultant(polchebyshev(n-1, 2, x/2), polchebyshev(m-1, 2, (4-x)/2)); \\ Michel Marcus, Apr 13 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A338832
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Number of spanning trees in the k_1 X ... X k_j grid graph, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.
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+10
1
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1, 1, 1, 4, 1, 15, 1, 384, 192, 56, 1, 31500, 1, 209, 2415, 42467328, 1, 49766400, 1, 2558976, 30305, 780, 1, 3500658000000, 100352, 2911, 8193540096000, 207746836, 1, 76752081000, 1, 20776019874734407680, 380160, 10864, 4140081, 242716067758080000000, 1
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listen;
history;
text;
internal format)
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OFFSET
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1,4
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COMMENTS
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a(n) > 1 precisely when n is composite.
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LINKS
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FORMULA
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a(n) = Product_{n_1=0..k_1-1, ..., n_j=0..k_j-1; not all n_i=0} Sum_{i=1..j} (2*(1 - cos(n_i*Pi/k_i))) / Product_{i=1..j} k_i, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.
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EXAMPLE
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The partition (2, 2, 1) has Heinz number 18 and the 3 X 3 X 2 grid graph has a(18) = 49766400 spanning trees.
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CROSSREFS
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2 X n grid: A001353(n) = a(2*prime(n-1))
3 X n grid: A006238(n) = a(3*prime(n-1))
4 X n grid: A003696(n) = a(5*prime(n-1))
5 X n grid: A003779(n) = a(7*prime(n-1))
6 X n grid: A139400(n) = a(11*prime(n-1))
7 X n grid: A334002(n) = a(13*prime(n-1))
8 X n grid: A334003(n) = a(17*prime(n-1))
9 X n grid: A334004(n) = a(19*prime(n-1))
10 X n grid: A334005(n) = a(23*prime(n-1))
n X n grid: A007341(n) = a(prime(n-1)^2)
m X n grid: A116469(m,n) = a(prime(m-1)*prime(n-1))
2 X 2 X n grid: A003753(n) = a(4*prime(n-1))
2 X n X n grid: A067518(n) = a(2*prime(n-1)^2)
n X n X n grid: A071763(n) = a(prime(n-1)^3)
2 X ... X 2 grid: A006237(n) = a(2^n)
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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