A182866 -id:A182866 - cp's OEIS frontend

A159299 Number of n-colorings of the 4 X 4 Sudoku graph.

0, 0, 0, 0, 288, 166560, 33539040, 2350746720, 75756999360, 1388552614848, 16744788486720, 146769785743680, 1002373493948640, 5606534724167520, 26640793339768608, 110556058012152480, 409297168707073920, 1374572399886053760, 4243833928227876480

Offset: 0

Alois P. Heinz, Apr 09 2009

The 4 X 4 Sudoku graph is a septic graph on 16 vertices and 56 edges. a(n) gives the number of 4 X 4 Sudoku solutions, if each of up to n numbers is allowed only once in every row, column and block.

			For n=4 colors one of the 288 possible colorings is given by this Sudoku:
  +---+---+
  |1 2|3 4|
  |4 3|2 1|
  +---+---+
  |3 1|4 2|
  |2 4|1 3|
  +---+---+ .

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Timme, Marc; van Bussel, Frank; Fliegner, Denny; Stolzenberg, Sebastian (2009) "Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions", New J. Phys. 11 023001, doi: 10.1088/1367-2630/11/2/023001.
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Wikipedia, Mathematics of Sudoku
Wikipedia, Sudoku
Wikipedia, Sudoku algorithms
Index entries for linear recurrences with constant coefficients, signature (17,-136,680, -2380,6188,-12376,19448,-24310,24310,-19448,12376,-6188,2380, -680,136,-17,1).

Cf. A107739, A182866.

a:= n-> n^16 -56*n^15 +1492*n^14 -25072*n^13 +296918*n^12 -2621552*n^11 +17795572*n^10 -94352168*n^9 +392779169*n^8 -1279118840*n^7 +3217758336*n^6 -6107865464*n^5 +8413745644*n^4 -7877463064*n^3 +4436831332*n^2 -1117762248*n: seq(a(n), n=0..20);

a(n) = n^16 -56*n^15 + ... (see Maple program).

G.f.: -96*x^4*(343316843*x^12 +4128584684*x^11 +20203233398*x^10 +50370257700*x^9 +68017469565*x^8 +50271571704*x^7 +20027437332*x^6 +4145554824*x^5 +419198325*x^4 +18781660*x^3 +320278*x^2 +1684*x +3)/ (x-1)^17. - Colin Barker, Aug 04 2012

A346008 Order of the full automorphism group of an n^2 X n^2 Sudoku puzzle.

Original entry on oeis.org

1, 128, 3359232, 126806761930752, 17832200896512000000000000, 20122639448358307421277388800000000000000, 346671850578027965617950152200042758191185920000000000000000, 158635147791426908154211087484339310324630213259159597497553256448000000000000000000, 3135383389315524601627656266493367412334920325664589642523187933340624422000766361791574835200000000000000000000

Offset: 1

Carl-Fredrik Nyberg Brodda, Shay Jordan, Jul 27 2021

nonn

a(n) is the order of the automorphism group of the n^2 X n^2 Sudoku graph (see A182866).

			For n=2, a(2) = 128 is the number of symmetries of a Shidoku puzzle.
For n=3, a(3) = 3359232 is the number of symmetries of standard 9 X 9 Sudoku puzzle.

Cf. A159299.

Join[{1},Table[2*n!^(2*n+2),{n,2,9}]] (* Stefano Spezia, Jul 27 2021 *)

M = matrix(n^4,n^4)
for i in [0..n^4-1]:
    for j in [0..n^4-1]:
        if i!=j:
            if i%n^2==j%n^2:
                M[i,j]=1
            if floor(i/n^2)==floor(j/n^2):
                M[i,j]=1
D = Graph(M, format='adjacency_matrix')
for col in [0..n-1]:
    for row in [0..n-1]:
        tl = n*col + n^3*row
        s = []
        for i in [0..n-1]:
            for j in [0..n-1]:
                s.append(tl + i + n^2*j)
        D.add_clique(s)
print(D.automorphism_group().order())

a(n) = 2*(n!)^(2n+2) for n > 1.

cp's OEIS Frontend

A159299 Number of n-colorings of the 4 X 4 Sudoku graph.

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