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.

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

Author

Pontus von Brömssen, Apr 15 2023

Keywords

• nonn

Comments

Also, a(n) is the maximum number of ways in which a set of integer-sided squares can tile an n X 2 rectangle, up to rotations and reflections.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..4771

Crossrefs

Row maxima of A102541.

Second column of A362258.

Cf. A001224, A073028, A361224 (rectangular pieces).

Programs

Maple

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

Formula

a(n) >= A073028(n)/2.