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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
| 0,3
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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).
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REFERENCES
| Sachkov, V. N. and Tarakanov, V. E., Combinatorics of Nonnegative Matrices. Translations of Mathematical Monographs, 213. American Mathematical Society, Providence, RI, 2002.
Wielandt, H. 1950. Unzerlegbare nicht negativen Matrizen, Math. Z. 52, 642-648.
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LINKS
| S. J. Leon, Linear Algebra with Applications: THE PERRON-FROBENIUS THEOREM
Index entries for sequences related to binary matrices
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FORMULA
| For asymptotics see Sachkov and Tarakanov.
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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} ]
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CROSSREFS
| Sequence in context: A030247 A139956 A016067 * A053527 A195632 A152504
Adjacent sequences: A070319 A070320 A070321 * A070323 A070324 A070325
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KEYWORD
| nonn,hard,more
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Aug 22 2003
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EXTENSIONS
| Wouter Meeussen (wouter.meeussen(AT)pandora.be) computed a(0) through a(4), Aug 22, 2003.
Jack Kennedy (kennedy(AT)oldnews.org) computed a(0) through a(5), Aug 22, 2003.
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