A265627 - OEIS

%I #19 Jul 25 2022 04:03:18

%S 2,10,498,65040,33554370,68718945018,562949953421058,

%T 18446744065119682560,2417851639229258080977408,

%U 1267650600228227149696920981450,2658455991569831745807614120560685058,22300745198530623141526273539119741048774160

%N Number of n X n "primitive" binary matrices.

%C A rectangular matrix is "primitive" in this sense if it cannot be expressed as a "tiling" of a single smaller matrix repeated in both directions.

%C Thus, for example, the 2 X 2 matrix with both rows equal to [1,0] is not primitive, since it can "tiled" by a single row.

%C This is the 2-dimensional generalization of A027375.

%F A general formula for the number of m X n "primitive" matrices over an alphabet of size k is Sum_{d|m, e|n} k^{m*n/(d*e)}*mu(d)*mu(e), where mu is the Möbius function.

%e We see a(2) = 10 since there are 16 possible 2 X 2 binary matrices, two are excluded because all their entries are the same, and four more are excluded because they are [[1,0],[1,0]] or a transpose or a negation.

%p with(numtheory):

%p prim := proc(k,m,n) option remember;

%p         dm := divisors(m);

%p         dn := divisors(n);

%p         s := 0;

%p         for d1 in dm do

%p                 for d2 in dn do

%p                         s := s+(k^(m*n/(d1*d2)))*mobius(d1)*mobius(d2);

%p                         od;

%p                 od;

%p         s;

%p         end:

%p seq(prim(2,n,n), n=1..40);

%t prim[k_, m_, n_] := prim[k, m, n] = Module[{s = 0},

%t Do[Do[s = s + (k^(m*n/(d1*d2)))*MoebiusMu[d1]*MoebiusMu[d2],

%t    {d1, Divisors[m]}], {d2, Divisors[n]}]; s];

%t a[n_] := prim[2, n, n]

%t Table[a[n], {n, 1, 12}] (* _Jean-François Alcover_, Jul 24 2022, after Maple code *)

%Y Cf. A027375.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, Dec 10 2015