primitive binary matrices?

[Meeussen Wouter (bkarnd)]

... and if you replace the condition of 
"strictly positive elements" by "strictly positive determinant,
then you get 
ID Number: A055165
URL:       http://www.research.att.com/projects/OEIS?Anum=A055165
Sequence:  1,6,174,22560,12514320,28836612000,270345669985440
Name:      Number of regular n X n matrices with rational entries equal to 0
or 1.
Comments:  All eigenvalues are nonzero.

intersting: the limitation to k<= n^2-2n+2 seems to work here too!

Table[it=Partition[#,n]&/@IntegerDigits[Range[0,-1+2^n^2],2,n^2]; Count
[it,(q_?MatrixQ) /; (Max@@Table[Det[MatrixPower[q,k] ],{k,1,n^2-2n+2}] )>0
]    ,{n,1,4}]
{1, 6, 174, 22560}

W.



-----Original Message-----
From: Meeussen Wouter (bkarnd)
[mailto:wouter.meeussen at vandemoortele.com]
Sent: vrijdag 22 augustus 2003 11:54
To: 'njas at research.att.com'; seqfan at ext.jussieu.fr
Subject: RE: primitive binary matrices?


in case I understood the quest:
is it {1, 3, 139, 25575} ?
(offset 1)

from 

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}]

Wouter.


-----Original Message-----
From: N. J. A. Sloane [mailto:njas at research.att.com]
Sent: vrijdag 22 augustus 2003 7:47
To: seqfan at ext.jussieu.fr
Cc: njas at research.att.com
Subject: primitive binary matrices?



Would someone work out a few terms of the following
sequence?

It is probably not in the OEIS and it should be!

a(n) = number of real n X n primitive (0,1)-matrices A.

Primitive means that for some k, every element of A^k is
strictly positive.

It is known that you don't need to check values
of k bigger than n^2 - 2n + 2
(this is a theorem of Wielandt, that if A
is primitive then k <= n^2 - 2n + 2)

References - see A002522.

The asymptotics are studied in the book

 Sachkov, V. N.; Tarakanov, V. E. Combinatorics of nonnegative matrices.
Translated from the 2000 Russian original by Valentin F. Kolchin.
Translations of Mathematical Monographs, 213. American Mathematical Society,
Providence, RI, 2002. x+269 pp. ISBN: 0-8218-2788-X (Reviewer: D. J.
Hartfiel) 15-02 (15A48)

NJAS


===============================
This email is confidential and intended solely for the use of the individual
to whom it is addressed. 
If you are not the intended recipient, be advised that you have received
this email in error and that any use, dissemination, forwarding, printing,
or copying of this email is strictly prohibited.
You are explicitly requested to notify the sender of this email that the
intended recipient was not reached.


===============================
This email is confidential and intended solely for the use of the individual to whom it is addressed. 
If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited.
You are explicitly requested to notify the sender of this email that the intended recipient was not reached.