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.
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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
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
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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))
A362259
Maximum number of ways in which a set of integer-sided squares can tile an n X n square, up to rotations and reflections.
Original entry on oeis.org
1, 1, 1, 1, 4, 20, 277, 7855, 487662
Offset: 0
A362262
Maximum number of ways in which a set of integer-sided squares can tile an n X 4 rectangle, up to rotations and reflections.
Original entry on oeis.org
1, 1, 2, 2, 4, 13, 33, 72, 204, 476, 1348, 3454, 9511, 25088, 68579, 186048, 503538, 1387536, 3732666, 10420102
Offset: 0
A362263
Maximum number of ways in which a set of integer-sided squares can tile an n X 5 rectangle, up to rotations and reflections.
Original entry on oeis.org
1, 1, 2, 4, 13, 20, 125, 403, 1438, 4718, 17700, 65811, 266345, 1036625, 3817494, 15399048
Offset: 0
Showing 1-5 of 5 results.
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