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A265627 Number of n X n "primitive" binary matrices. 9
2, 10, 498, 65040, 33554370, 68718945018, 562949953421058, 18446744065119682560, 2417851639229258080977408, 1267650600228227149696920981450, 2658455991569831745807614120560685058, 22300745198530623141526273539119741048774160 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

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.

This is the 2-dimensional generalization of A027375.

LINKS

Table of n, a(n) for n=1..12.

FORMULA

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.

EXAMPLE

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.

MAPLE

with(numtheory):

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

        dm := divisors(m);

        dn := divisors(n);

        s := 0;

        for d1 in dm do

                for d2 in dn do

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

                        od;

                od;

        s;

        end:

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

CROSSREFS

Cf. A027375.

Sequence in context: A207140 A059723 A334286 * A112449 A011824 A064300

Adjacent sequences:  A265624 A265625 A265626 * A265628 A265629 A265630

KEYWORD

nonn

AUTHOR

Jeffrey Shallit, Dec 10 2015

STATUS

approved

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Last modified May 26 02:56 EDT 2020. Contains 334613 sequences. (Running on oeis4.)