[seqfan] Re: Dissecting a square into squares, unfinished business

Robert G. Wilson v rgwv at rgwv.com
Sun Apr 14 18:18:10 CEST 2013


Et al,

	Neil just informed us that A034295 and A140108 are indeed the same
and have been merged into the former.

	This note is just to point out why this sequence is marked 'hard'.
The example for a(5) being equal to 11 as opposed to just partitioning
integers into square parts. see https://oeis.org/A001156

	Here are the 19 partitions (of A001156) using the format of the
example in A034295. I have placed an asterisk in front of those 8 sets which
cannot be represented  using planar figures.

Sincerely, Bob.

  1. 25 (1x1)
  2.  1 (2x2) + 21 (1x1)
  3.  2 (2x2) + 17 (1x1)
  4.  3 (2x2) + 13 (1x1)
  5.  4 (2x2) +  9 (1x1)
* 6.  5 (2x2) +  5 (1x1)
* 7.  6 (2x2) +  1 (1x1)
  8.  1 (3x3) + 16 (1x1)
  9.  1 (3x3) +  1 (2x2) + 12 (1x1)
 10.  1 (3x3) +  2 (2x2) +  8 (1x1)
 11.  1 (3x3) +  3 (2x2) +  4 (1x1)
*12.  1 (3x3) +  4 (2x2)
*13.  2 (3x3) +  7 (1x1)
*14.  2 (3x3) +  1 (2x2) +  3 (1x1)
 15.  1 (4x4) +  9 (1x1)
*16.  1 (4x4) +  1 (2x2) +  5 (1x1)
*17.  1 (4x4) +  2 (2x2) +  1 (1x1)
*18.  1 (4x4) +  1 (3x3)
 19.  1 (5x5)

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Neil Sloane
Sent: Friday, April 12, 2013 11:37 PM
To: Sequence Fanatics Discussion list
Subject: [seqfan] Dissecting a square into squares, unfinished business

Dear Seq Fans, Early this morning Paola Lava pointed out that there were two
versions of A034295 in the OEIS. I merged them and now there is only one.

But this looks at how many ways there are to dissect a square into squares,
KEEPING TRACK ONLY OF THE NUMBERS OF PARTS OF EACH SIZE, and ignoring the
geometry.

On the other hand, A045846 gives the number of ways when you do look at the
geometry. E.g. a(3) = 6: you can have a 3x3 square, or nine 1x1 squares, or
one 2x2 square and 5 1x1 squares where the 2x2 square can be in any of the 4
corners: total is A045846(3)=6.

But if we take the latter sequence and say that two dissections are
equivalent if a rotation/reflection takes one into the other, the number of
inequivalent dissections is (for n=1,2,3,4)
1,2,3,13
and I think this sequence is missing from the the OEIS.

I can send anyone who is interested a drawing of the first 4 terms (we are
not allowed attachments on this mailing list).

How does it continue?

Neil

--
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com

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