A360999 -id:A360999 - cp's OEIS frontend

A360998 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling), 1 <= k <= n.

1, 2, 2, 2, 3, 2, 3, 4, 4, 3, 2, 3, 3, 4, 2, 4, 6, 5, 7, 5, 4

Offset: 1

Pontus von Brömssen, Feb 28 2023

It seems that each solution consists of n*k/(r*s) copies of an r X s piece (arranged in a simple grid, all pieces oriented in the same way), where r is a divisor of n, s is a divisor of k, and either r = s or r is not a divisor of k or s is not a divisor of n. If this is true, T(n,k) <= d(n)*d(k) - d(m)*(d(m)-1), where d = A000005 is the divisor count function and m = gcd(n,k). Equality does not always hold; for (n,k) = (3,2), for example, (r,s) = (1,2) satisfies the condition, but three 1 X 2 pieces can tile the 3 X 2 rectangle in more than one way.

Is d(n)*d(k) - T(n,k) eventually periodic in n for each k?

			Triangle begins:
  n\k|  1  2  3  4  5  6
  ---+------------------
  1  |  1
  2  |  2  2
  3  |  2  3  2
  4  |  3  4  4  3
  5  |  2  3  3  4  2
  6  |  4  6  5  7  5  4
The T(4,3) = 4 nonrearrangeable tilings of the 4 X 3 rectangle are:
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +           +   +           +   +   +   +   +   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +           +   +---+---+---+   +   +   +   +   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +           +   +           +   +   +   +   +   +---+---+---+
  |           |   |           |   |   |   |   |   |   |   |   |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+

Columns: A000005 (k = 1), A360999 (k = 2), A361000 (k = 3).

Cf. A083039, A360629, A361001.

T(n,1) = d(n) = A000005(n).
T(n,2) = A360999(n) = 2*d(n) - 1 - [n even] for n >= 2.
T(n,3) = A361000(n) = 2*d(n) - A083039(n) for n >= 3.
It appears that T(n,4) = 3*d(n) - 2 - 2*[n even] - [n divisible by 3] - 2*[n divisible by 4] for n >= 4.
It appears that T(n,n) = d(n). (It is easy to see that T(n,n) >= d(n).)

A361004 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling).

Original entry on oeis.org

2, 4, 7, 11, 14, 23, 22, 32, 37, 42, 39, 69, 50, 64

Offset: 1

Pontus von Brömssen, Feb 28 2023

Second column of A361001.

Cf. A360631, A360999.

cp's OEIS Frontend

A360998 Triangle read by rows: T(n,k) is the number of tilings of an n X k rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (including rotations and reflections of the original tiling), 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A361004 Number of tilings of an n X 2 rectangle by integer-sided rectangular pieces that cannot be rearranged to produce a different tiling of the rectangle (except rotations and reflections of the original tiling).

Views

Author

Keywords

Crossrefs