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 A070322 Number of primitive n X n real (0,1)-matrices. 1
 1, 1, 3, 139, 25575, 18077431 (list; graph; refs; listen; history; text; internal format)
 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 Nicholas R. Beaton, Walks obeying two-step rules on the square lattice: full, half and quarter planes, arXiv:2010.06955 [math.CO], 2020. S. J. Leon, Linear Algebra with Applications: the Perron-Frobenius Theorem [Cached copy at the Wayback Machine] Helmut Wielandt, Unzerlegbare, nicht negative Matrizen, Math. Z. 52 (1950), 642-648. 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^2-2n+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

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)