semimagic squares: 1, 0, 9, ...

[זקיר סעידוב - ד\"ר/Zakir Seidov Ph.D.]

Eric, and all SeqFans,

how many are there 
3x3 squares modulo rotation/reflection 
with 1, 2, 3, ..., 9 distinct digits?

with 1 digit we have n=3  distinct squares (from 9 without modulo rot/ref):
1 . .    . 1 .   . . .
. . .    . . .   . 1 .
. . .    . . .   . . .
======   =====   =====
  1-1      1-2    1-3

with 2 digits we have n=12 (from  9*8=72 without modulo rot/ref):
1 2 .    1 . 2   1 . .   1 . .    1 . .
. . .    . . .   . 2 .   . . 2    . . .
. . .    . . .   . . .   . . .    . . 2
=====    =====   =====   =====    =====
 2-1      2-2     2-3     2-4      2-5


. 1 2    . 1 .   . 1 .   . 1 .     . 1 .
. . .    . 2 .   . . 2   . . .     . . .
. . .    . . .   . . .   . 2 .     . . 2
=====    =====   =====   =====     =====
 2-6      2-7     2-8     2-9       2-10

. 2 .    . . 2
. 1 .    . 1 .
. . .    . . .
======   =====   
 2-11     2-12


also:

with 3 digits we have n=66            (from      72*7=504 without modulo rot/ref)
with 4 digits we have n=378           (from     504*6=3024 without modulo rot/ref)

now, because for 4 digits there are no symmetrical squares,
the next figures are trivial:

with 5 digits we have n=378*5=1890    (from    3024*5=15120 without modulo rot/ref)
with 6 digits we have n=1890*4=7560   (from   15120*4=60480 without modulo rot/ref)
with 7 digits we have n=7560*3=22680  (from   60480*3=181440 without modulo rot/ref)
with 8 digits we have n=22680*2=45360 (from  181440*2=362880 without modulo rot/ref)
with 9 digits we have n=45360*1=45360 (from        9!=362880 without modulo rot/ref)

Are you agree with these figures?
How to generate all these 
say 45360 distinct squares (for say 8/9 digits)?

Neil,
are
3,12,66,378,1890,7560,22680,45360,45360 
OK for OEIS?

sorry for lengthy post...
zak

%%%%%%%% 

-----Original Message-----
From: Eric W. Weisstein [mailto:eww at wolfram.com]
Sent: Thursday, August 07, 2003 2:57 PM
To: Sequence Fans Mailing List
Subject: semimagic squares: 1, 0, 9, ...


There are nine *semimagic* (cf. A006052) squares of order 3 modulo
rotation/reflection:

{{{1, 5, 9}, {6, 7, 2}, {8, 3, 4}},
{{1, 5, 9}, {8, 3, 4}, {6, 7, 2}},
{{1, 6, 8}, {9, 2, 4}, {5, 7, 3}},
{{1, 8, 6}, {9, 4, 2}, {5, 3, 7}},
{{2, 4, 9}, {6, 8, 1}, {7, 3, 5}},
{{2, 6, 7}, {9, 1, 5}, {4, 8, 3}},
{{2, 7, 6}, {9, 5, 1}, {4, 3, 8}},
{{3, 4, 8}, {5, 9, 1}, {7, 2, 6}},
{{3, 7, 5}, {8, 6, 1}, {4, 2, 9}}}

How many are there for higher orders?  Any takers?

Cheers,
-E