A360630 Number of sets of integer-sided rectangular pieces that can tile an n X n square.

1, 4, 21, 192, 2035, 27407, 399618

Offset

1,2

Links

Table of n, a(n) for n=1..7.

Krystian Gajdzica, Robin Visser, and Maciej Zakarczemny, Rectangle partitions generalizing integer partitions, Ann. Comb. (2026). See also arXiv:2509.20495 [math.CO], 2025.

Example

The a(3) = 21 possible sets of pieces that can tile a 3 X 3 square are given in the table below. (Each column on the right gives a set of pieces.)
.
  length X width |            number of pieces
  ---------------+------------------------------------------
       3 X 3     | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
       2 X 3     | 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
       2 X 2     | 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0
       1 X 3     | 0 1 0 0 1 1 0 0 0 3 2 2 1 1 1 1 0 0 0 0 0
       1 X 2     | 0 0 1 0 1 0 2 1 0 0 1 0 3 2 1 0 4 3 2 1 0
       1 X 1     | 0 0 1 3 0 2 1 3 5 0 1 3 0 2 4 6 1 3 5 7 9

Crossrefs

Main diagonal of A360629.

Cf. A034295 (square pieces).

Sequence in context: A065527 A267988 A387066 * A041667 A377828 A286883

Adjacent sequences: A360627 A360628 A360629 * A360631 A360632 A360633

Keyword

nonn,more

Author

Pontus von Brömssen, Feb 14 2023

Extensions

a(7) from Robin Visser, May 09 2025

Status

approved