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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+---+---+---+ +---+---+---+
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.
A361218
Maximum number of ways in which a set of integer-sided rectangular pieces can tile an n X 2 rectangle.
Original entry on oeis.org
1, 4, 11, 29, 94, 263, 968, 3416, 11520, 41912, 136972, 481388, 1743784, 6275886, 23615432, 93819128, 368019576, 1367900808, 5403282616, 19831367476, 76031433360, 300581321056, 1143307393600, 4542840116352, 17001097572544, 65314285778004, 246695766031432
Offset: 1
The following table shows the sets of pieces that give the maximum number of tilings for n <= 27. The solutions are unique except for n = 1.
\ Number of pieces of size
n \ 1 X 1 | 1 X 2 | 1 X 3 | 1 X 4
----+-------+-------+-------+------
1 | 2 | 0 | 0 | 0
1 | 0 | 1 | 0 | 0
2 | 2 | 1 | 0 | 0
3 | 2 | 2 | 0 | 0
4 | 4 | 2 | 0 | 0
5 | 4 | 3 | 0 | 0
6 | 4 | 4 | 0 | 0
7 | 5 | 3 | 1 | 0
8 | 5 | 4 | 1 | 0
9 | 7 | 4 | 1 | 0
10 | 7 | 5 | 1 | 0
11 | 7 | 6 | 1 | 0
12 | 9 | 6 | 1 | 0
13 | 8 | 6 | 2 | 0
14 | 10 | 6 | 2 | 0
15 | 10 | 7 | 2 | 0
16 | 10 | 6 | 2 | 1
17 | 10 | 7 | 2 | 1
18 | 12 | 7 | 2 | 1
19 | 12 | 8 | 2 | 1
20 | 12 | 9 | 2 | 1
21 | 13 | 8 | 3 | 1
22 | 13 | 9 | 3 | 1
23 | 15 | 9 | 3 | 1
24 | 15 | 10 | 3 | 1
25 | 15 | 11 | 3 | 1
26 | 17 | 11 | 3 | 1
27 | 17 | 12 | 3 | 1
A362260
Maximum over 0 <= k <= n/2 of the number of permutations of two symbols occurring k and n-2*k times, respectively, where a permutation and its reversal are counted only once.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 4, 6, 9, 12, 19, 28, 44, 66, 110, 170, 255, 396, 651, 1001, 1519, 2520, 4032, 6216, 9752, 15912, 25236, 38760, 63090, 101850, 160050, 248710, 408760, 653752, 1021735, 1634776, 2656511, 4218786, 6562556, 10737090, 17299646, 27313650, 43249115
Offset: 0
For n = 8, the maximum a(8) = 9 is obtained for k = 2. The corresponding permutations of 2 2's and 4 1's are 221111, 212111, 211211, 211121, 211112, 122111, 121211, 121121, and 112211.
-
f:= proc(n) local k, v, m,w;
m:= 0:
for k from 0 to n/2 do
v:= binomial(n-k,k);
if n:: even and k::even then w:= binomial((n-k)/2,k/2)
elif (n-k)::odd then w:=binomial((n-k-1)/2, floor(k/2))
else w:= 0
fi;
m:= max(m,(v+w)/2);
od;
m
end proc:
map(f, [$0..50]); # Robert Israel, Oct 25 2023
A361426
Maximum difficulty level (see A361424 for the definition) for tiling an n X 2 rectangle with a set of integer-sided rectangles, rounded down to the nearest integer.
Original entry on oeis.org
2, 2, 6, 12, 16, 48, 53, 120, 320, 280, 1120, 2240, 2986, 8960, 17920, 26880, 53760, 107520, 134400, 268800, 537600, 591360, 1182720, 2365440, 2956800, 5677056, 11354112
Offset: 1
The following table shows all sets of pieces that give the maximum (n,2)-tiling difficulty level up to n = 27.
\ Number of pieces of size
n \ 1X1 | 1X2 | 1X3 | 1X4 | 1X5 | 1X7 | 2X2 | 2X3
----+-----+-----+-----+-----+-----+-----+-----+----
1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0
2 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0
3 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0
4 | 0 | 2 | 0 | 1 | 0 | 0 | 0 | 0
4 | 0 | 1 | 2 | 0 | 0 | 0 | 0 | 0
5 | 1 | 2 | 0 | 0 | 1 | 0 | 0 | 0
5 | 0 | 3 | 0 | 1 | 0 | 0 | 0 | 0
6 | 0 | 1 | 2 | 1 | 0 | 0 | 0 | 0
7 | 0 | 1 | 4 | 0 | 0 | 0 | 0 | 0
8 | 2 | 0 | 2 | 1 | 0 | 0 | 1 | 0
8 | 0 | 1 | 2 | 1 | 0 | 0 | 1 | 0
9 | 1 | 0 | 3 | 2 | 0 | 0 | 0 | 0
10 | 2 | 0 | 2 | 1 | 0 | 0 | 2 | 0
11 | 1 | 0 | 3 | 2 | 0 | 0 | 1 | 0
12 | 1 | 0 | 3 | 2 | 0 | 0 | 0 | 1
12 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 0
13 | 1 | 0 | 3 | 2 | 0 | 0 | 2 | 0
14 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 0
15 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 1
16 | 0 | 0 | 4 | 3 | 0 | 0 | 2 | 0
17 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 1
18 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 2
19 | 0 | 0 | 4 | 3 | 0 | 0 | 2 | 1
20 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 2
21 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 3
22 | 0 | 0 | 5 | 2 | 0 | 1 | 2 | 1
22 | 0 | 0 | 5 | 0 | 3 | 0 | 2 | 1
22 | 0 | 0 | 4 | 3 | 0 | 0 | 2 | 2
23 | 0 | 0 | 5 | 2 | 0 | 1 | 1 | 2
23 | 0 | 0 | 5 | 0 | 3 | 0 | 1 | 2
23 | 0 | 0 | 4 | 3 | 0 | 0 | 1 | 3
24 | 0 | 0 | 5 | 2 | 0 | 1 | 0 | 3
24 | 0 | 0 | 5 | 0 | 3 | 0 | 0 | 3
24 | 0 | 0 | 4 | 3 | 0 | 0 | 0 | 4
25 | 0 | 0 | 3 | 4 | 0 | 1 | 0 | 3
26 | 0 | 0 | 5 | 2 | 0 | 1 | 1 | 3
26 | 0 | 0 | 5 | 0 | 3 | 0 | 1 | 3
27 | 0 | 0 | 5 | 2 | 0 | 1 | 0 | 4
27 | 0 | 0 | 5 | 0 | 3 | 0 | 0 | 4
Showing 1-4 of 4 results.
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