

A070322


Number of primitive n X n real (0,1)matrices.


1




OFFSET

0,3


COMMENTS

An n X n nonnegative matrix A is primitive iff every element of A^k is > 0 for some power k. If A is primitive then the power which should have all positive entries is <= n^2  2n + 2 (Wielandt).


REFERENCES

Sachkov, V. N. and Tarakanov, V. E., Combinatorics of Nonnegative Matrices. Translations of Mathematical Monographs, 213. American Mathematical Society, Providence, RI, 2002.


LINKS

Table of n, a(n) for n=0..5.
Nicholas R. Beaton, Walks obeying twostep rules on the square lattice: full, half and quarter planes, arXiv:2010.06955 [math.CO], 2020.
S. J. Leon, Linear Algebra with Applications: the PerronFrobenius Theorem [Cached copy at the Wayback Machine]
Helmut Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642648.
Index entries for sequences related to binary matrices


FORMULA

For asymptotics see Sachkov and Tarakanov.


MATHEMATICA

Table[ it=Partition[ #, n ]&/@IntegerDigits[ Range[ 0, 1+2^n^2 ], 2, n^2 ]; Count [ it, (q_?MatrixQ) /; (Max@@Table[ Min@@Flatten[ MatrixPower[ q, k ] ], {k, 1, n^22n+2} ] )>0 ], {n, 1, 4} ]


CROSSREFS

Sequence in context: A030247 A139956 A236193 * A053527 A195632 A152504
Adjacent sequences: A070319 A070320 A070321 * A070323 A070324 A070325


KEYWORD

nonn,hard,more


AUTHOR

N. J. A. Sloane, Aug 22 2003


EXTENSIONS

Wouter Meeussen computed a(0) through a(4), Aug 22 2003
I. J. Kennedy computed a(0) through a(5), Aug 22 2003


STATUS

approved



