A319926 Isomer counts of compound perfect squared squares.

4, 7, 8, 11, 12, 14, 16

Offset: 1

Author

Stuart E Anderson, Oct 01 2018

Keywords

• nonn
• more

Comments

The isomer count of a compound perfect squared square (CPSS) is the number of ways its squared subrectangle and constituent squares can be arranged, up to symmetry of the CPSS. A squared square is perfect if none of its constituent squares are the same size. A squared square is compound if it contains a smaller squared subrectangle. Note that the squared subrectangle can be a squared square. Specific concrete examples of CPSSs with isomer counts under 100 of 4, 7, 8, 11, 12, 16, 19, 20, 23, 24, 28, 31, 32, 35, 36, 39, 40, 47, 48, 56, 60, 63, 64, 68, 72, 76, 80, 88 and 96 exist. Geometric constructions based on a suitable pair of perfect squared rectangles each with up to 4 isomers suggests additional isomer counts up to 100 of 14, 21, 22, 33, 42, 44, 66 and 99, but no actual examples are known. As the number of squares in a squared square - the order - increases new arrangements appear. It is conjectured that expected CPSS subrectangle isomer arrangements will eventually appear if the order is high enough.

The term a(6)=14 is based on a theoretical construction, not on known or existing CPSSs. These terms have been included to distinguish the sequence from others. Considering all the ways two or more subrectangles can be arranged within a CPSS it does not appear possible for a CPSS with 5, 6, 9, 10 or 13 isomers to exist but even this much has not been proved.

Examples

			a(1) = 4, because the compound perfect squares of order 24 comprise the square with side 175 and Bouwkamp code (81,56,38) (18,20) (55,16,3) (1,5,14) (4) (9) (39) (51,30) (29,31,64) (43,8) (35,2) (33) as well as three others from the other symmetries of the order-13 111 X 94 squared subrectangle. See MathWorld link for an explanation of Bouwkamp code.
		

Links

• Stuart E Anderson, Compound Perfect Squared Squares of the Order Twenties, 2013; arXiv:1303.0599 [math.CO], 2013.
• Stuart E Anderson, Compound Perfect squared Squares
• Stuart E Anderson, 61 page PDF document with images of all the isomers of CPSSs with isomer counts of 4, 7, 8, 11, 12, 16, 19, 20, 23, 24, 28, 31, 32, 35, 36, 39, 40, 47, 48, 56, 60, 63, 64, 68, 72, 76, 80, 88, 96.
• A. J. W. Duijvestijn, P. J. Federico and P. Leeuw, Compound perfect squares, Amer. Math. Monthly 89 (1982), 15-32.
• N. D. Kazarinoff and R. Weitzenkamp, On the existence of compound perfect squared squares of small order, J. Combin. Theory Ser. B 14 (1973), 163-179.
• Eric Weisstein's World of Mathematics, Perfect Square Dissection
• Jim Williams, programs to generate and count compound perfect squared squares and their isomers

Crossrefs

Cf. A181340, A217155.