A361221
Triangle read by rows: T(n,k) is the maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X k rectangle, up to rotations and reflections.
Original entry on oeis.org
1, 1, 1, 1, 5, 8, 2, 12, 95, 719, 2, 31, 682, 20600, 315107
Offset: 1
Triangle begins:
n\k| 1 2 3 4 5
---+------------------------
1 | 1
2 | 1 1
3 | 1 5 8
4 | 2 12 95 719
5 | 2 31 682 20600 315107
A 3 X 3 square can be tiled by one 1 X 3 piece, two 1 X 2 pieces and two 1 X 1 pieces in the following 8 ways:
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+---+---+ + +---+---+---+ +---+---+---+
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+---+---+---+ +---+---+---+ +---+---+---+
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+---+---+---+ +---+---+---+ +---+---+---+
.
+---+---+---+ +---+---+---+ +---+---+---+
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+ + +---+ + +---+ + + +---+---+
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+---+---+---+ +---+---+---+ +---+---+---+
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+---+---+---+ +---+---+---+ +---+---+---+
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+---+---+---+ +---+---+---+
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This is the maximum for a 3 X 3 square, so T(3,3) = 8. There is one other set of pieces that also can tile the 3 X 3 square in 8 ways: three 1 X 2 pieces and three 1 X 1 pieces (see illustration in A361216).
The following table shows all sets of pieces that give the maximum number of tilings for 1 <= k <= n <= 5:
\ Number of pieces of size
(n,k)\ 1 X 1 | 1 X 2 | 1 X 3 | 2 X 2
------+-------+-------+-------+------
(1,1) | 1 | 0 | 0 | 0
(2,1) | 2 | 0 | 0 | 0
(2,1) | 0 | 1 | 0 | 0
(2,2) | 4 | 0 | 0 | 0
(2,2) | 2 | 1 | 0 | 0
(2,2) | 0 | 2 | 0 | 0
(2,2) | 0 | 0 | 0 | 1
(3,1) | 3 | 0 | 0 | 0
(3,1) | 1 | 1 | 0 | 0
(3,1) | 0 | 0 | 1 | 0
(3,2) | 2 | 2 | 0 | 0
(3,3) | 3 | 3 | 0 | 0
(3,3) | 2 | 2 | 1 | 0
(4,1) | 2 | 1 | 0 | 0
(4,2) | 4 | 2 | 0 | 0
(4,3) | 3 | 3 | 1 | 0
(4,4) | 5 | 4 | 1 | 0
(5,1) | 3 | 1 | 0 | 0
(5,1) | 2 | 0 | 1 | 0
(5,1) | 1 | 2 | 0 | 0
(5,2) | 4 | 3 | 0 | 0
(5,3) | 4 | 4 | 1 | 0
(5,4) | 7 | 5 | 1 | 0
(5,5) | 7 | 6 | 2 | 0
It seems that all optimal solutions for A361216 are also optimal here, but occasionally there are other optimal solutions, e.g. for n = k = 3.
A361219
Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 3 rectangle.
Original entry on oeis.org
2, 11, 56, 370, 2666, 19126, 134902, 1026667, 8049132, 60996816, 450456500, 3427769018, 27127841200, 211563038980, 1837421211974, 15474223886906
Offset: 1
The following table shows the sets of pieces that give the maximum number of tilings for n <= 16. All these sets are unique.
\ Number of pieces of size
n \ 1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
----+-------+-------+-------+------
1 | 1 | 1 | 0 | 0
2 | 2 | 2 | 0 | 0
3 | 3 | 3 | 0 | 0
4 | 3 | 3 | 1 | 0
5 | 4 | 4 | 1 | 0
6 | 7 | 4 | 1 | 0
7 | 8 | 5 | 1 | 0
8 | 8 | 5 | 2 | 0
9 | 9 | 6 | 2 | 0
10 | 10 | 7 | 2 | 0
11 | 11 | 8 | 2 | 0
12 | 11 | 8 | 3 | 0
13 | 12 | 9 | 3 | 0
14 | 13 | 8 | 3 | 1
15 | 14 | 9 | 3 | 1
16 | 15 | 10 | 3 | 1
A362261
Maximum number of ways in which a set of integer-sided squares can tile an n X 3 rectangle, up to rotations and reflections.
Original entry on oeis.org
1, 1, 1, 1, 2, 4, 8, 12, 22, 40, 73, 146, 292, 560, 1120, 2532, 5040, 10080, 22176, 44352, 88704, 192272, 384384, 768768, 1647360, 3294720, 6589440, 14003120, 28006240, 56012480, 126028080, 266053680, 532107360, 1182438400, 2483130720, 4966261440, 10925775168
Offset: 0
-
from math import comb
def F(i,j,k):
# total number of tilings using i, j, and 2*j+3*k squares of side lengths 3, 2, and 1, respectively
return comb(i+j+k,i)*comb(j+k,j)*2**j
def F0(i,j,k):
# number of inequivalent tilings
x = F(i,j,k)
if j == 0: x += comb(i+k,i) # horizontal line of symmetry
if i%2+j%2+k%2 <= 1: x += 2*F(i//2,j//2,k//2) # vertical line of symmetry or rotational symmetry
return x//4
def A362261(n):
return max(F0(i,j,n-3*i-2*j) for i in range(n//3+1) for j in range((n-3*i)//2+1))
A361427
Maximum difficulty level (see A361424 for the definition) for tiling an n X 3 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.
Original entry on oeis.org
2, 6, 8, 48, 80, 480, 960, 1920, 3360, 13440, 20160, 60480, 80640, 201600, 967680, 1612800
Offset: 1
The following table shows all sets of pieces that give the maximum (n,3)-tiling difficulty level up to n = 16.
\ Number of pieces of size
n \ 1X1|1X2|1X3|1X4|1X5|1X6|1X7|1X8|1X9|1X10|1X12|2X2|2X3|2X4|2X5
----+---+---+---+---+---+---+---+---+---+----+----+---+---+---+---
1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
3 | 1 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
4 | 0 | 1 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
5 | 0 | 3 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
5 | 0 | 1 | 3 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
6 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0
7 | 0 | 3 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0
7 | 0 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0
8 | 0 | 3 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0
9 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0
9 | 2 | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0
9 | 0 | 4 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0
9 | 0 | 3 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 2 | 0 | 0 | 0
10 | 3 | 0 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1
11 | 0 | 2 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0
12 | 3 | 0 | 0 | 2 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 1
13 | 1 | 0 | 1 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1
13 | 0 | 2 | 0 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0
14 | 4 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 0 | 2 | 0
14 | 3 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 3 | 0 | 0 | 1
14 | 2 | 0 | 0 | 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0
15 | 1 | 0 | 0 | 3 | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1
16 | 3 | 0 | 0 | 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 1
Showing 1-4 of 4 results.
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