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A070322
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Number of primitive n X n real (0,1)-matrices.
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1
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OFFSET
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0,3
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COMMENTS
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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).
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REFERENCES
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Sachkov, V. N. and Tarakanov, V. E., Combinatorics of Nonnegative Matrices. Translations of Mathematical Monographs, 213. American Mathematical Society, Providence, RI, 2002.
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LINKS
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Table of n, a(n) for n=0..5.
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.
Index entries for sequences related to binary matrices
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FORMULA
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For asymptotics see Sachkov and Tarakanov.
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MATHEMATICA
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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} ]
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CROSSREFS
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Sequence in context: A030247 A139956 A236193 * A053527 A195632 A152504
Adjacent sequences: A070319 A070320 A070321 * A070323 A070324 A070325
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KEYWORD
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nonn,hard,more
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AUTHOR
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N. J. A. Sloane, Aug 22 2003
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EXTENSIONS
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Wouter Meeussen computed a(0) through a(4), Aug 22 2003
I. J. Kennedy computed a(0) through a(5), Aug 22 2003
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STATUS
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approved
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