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A054252 Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.

%I A054252 #57 Nov 24 2018 10:53:27
%S A054252 1,1,1,1,1,2,1,1,1,3,8,16,23,23,16,8,3,1,1,3,21,77,252,567,1051,1465,
%T A054252 1674,1465,1051,567,252,77,21,3,1,1,6,49,319,1666,6814,22475,60645,
%U A054252 136080,256585,410170,559014,652048,652048,559014,410170,256585,136080
%N A054252 Triangle T(n,k) of n X n binary matrices with k=0..n^2 ones under action of dihedral group of the square D_4.
%C A054252 From _Geoffrey Critzer_, Feb 19 2013: (Start)
%C A054252 Cycle indices for n=2,3,4,5 respectively are:
%C A054252 (1/8)(s[1]^4 + 2*s[1]^2*s[2] + 3*s[2]^2 + 2*s[4]).
%C A054252 (1/8)(s[1]^9 + 4*s[1]^3*s[2]^3 + s[1]s[2]^4 + 2*s[1]*s[4]^2).
%C A054252 (1/8)(s[1]^16 + 2*s[1]^4*s[2]^6 + 2*s[4]^4 + 3*s[2]^8).
%C A054252 (1/8)(s[1]^25 + 4*s[1]^5*s[2]^10 + 2*s[1]*s[4]^6 + s[1]*s[2]^12).
%C A054252 (End)
%C A054252 Also the number of equivalence classes of ways of placing k 1 X 1 tiles in an n X n square under all symmetry operations of the square. - _Christopher Hunt Gribble_, Feb 17 2014
%C A054252 From _Wolfdieter Lang_, Oct 03 2016: (Start)
%C A054252 The cycle index G(n) for a square n X n grid with squares coming in two colors with k squares of one color is for the D_4 group (with 8 elements R(90)^j, S R(90)^j, j=0..3)
%C A054252   (s[1]^(n^2) + s[2]^(n^2/2) +2*s[4]^(n^2/4))/8 + (s[2]^(n^2/2) + s[1]^n*s[2]^((n^2-n)/2))/4 if n is even,
%C A054252   s[1]*((s[1]^(n^2-1) + s[2]^((n^2-1)/2) + 2*s[4]^((n^2-1)/4))/8) + s[1]^n*s[2]^(n*(n-1)/2)/2 if n is odd.
%C A054252 See the above comment by _Geoffrey Critzer_ for n=2..5.
%C A054252 The figure counting series is c(x) = 1 + x for coloring, say black and white.
%C A054252 Therefore the counting series is C(n,x) = G(n) with substitution s[2^j] = c(x^(2*j)) = 1 + x^(2^j) for j=0,1,2. Row n gives the coefficients of C(n,x) in rising (or falling) order.  This follows from Pólya's counting theorem.  See the Harary-Palmer reference, p. 42, eq. (2.4.6), and eq. (2.2.11) with n=4 on p. 37 for the cycle index of D_4.
%C A054252 (End)
%D A054252 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 42, (2.4.6), p. 37, (2.2.11).
%H A054252 Heinrich Ludwig, <a href="/A054252/b054252.txt">Rows n = 0..16, flattened</a>
%H A054252 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%e A054252 T(3,2) = 8 because there are 8 nonisomorphic 3 X 3 binary matrices with two ones under action of D_4:
%e A054252   [0 0 0] [0 0 0] [0 0 0] [0 0 0]
%e A054252   [0 0 0] [0 0 0] [0 0 1] [0 0 1]
%e A054252   [0 1 1] [1 0 1] [0 1 0] [1 0 0]
%e A054252 ---------------------------------
%e A054252   [0 0 0] [0 0 0] [0 0 0] [0 0 1]
%e A054252   [0 1 0] [0 1 0] [1 0 1] [0 0 0]
%e A054252   [0 0 1] [0 1 0] [0 0 0] [1 0 0]
%e A054252 Triangle T(n,k) begins:
%e A054252 1;
%e A054252 1, 1;
%e A054252 1, 1, 2,  1,  1;
%e A054252 1, 3, 8, 16, 23, 23, 16, 8, 3, 1;
%t A054252 (* As a triangle *) Prepend[Prepend[Table[CoefficientList[CycleIndexPolynomial[
%t A054252 GraphData[{"Grid", {n, n}}, "AutomorphismGroup"],Table[Subscript[s, i], {i, 1, 4}]] /. Table[Subscript[s, i] -> 1 + x^i, {i, 1, 4}], x], {n, 2, 10}], {1, 1}], {1}] // Grid (* _Geoffrey Critzer_, Aug 09 2016 *)
%o A054252 (Sage)
%o A054252 def T(n, k):
%o A054252     if n == 0 or k == 0 or k == n*n:
%o A054252         return 1
%o A054252     grid = graphs.Grid2dGraph(n, n)
%o A054252     m = grid.automorphism_group().cycle_index().expand(2, 'b, w')
%o A054252     b, w = m.variables()
%o A054252     return m.coefficient({b: k, w: n*n-k})
%o A054252 [T(n, k) for n in range(6) for k in range(n*n + 1)] # _Freddy Barrera_, Nov 23 2018
%Y A054252 Cf. A014409, A019318, A054247 (row sums), A054772.
%K A054252 easy,nonn,tabf
%O A054252 0,6
%A A054252 _Vladeta Jovovic_, May 04 2000