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

0,5

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.

Links

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

Formula

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

Example

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.

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

Crossrefs

Row maxima of A102541.

Second column of A362258.

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

Sequence in context: A035564 A240065 A237755 * A306730 A209603 A192684

Adjacent sequences: A362257 A362258 A362259 * A362261 A362262 A362263

Keyword

nonn

Author

Pontus von Brömssen, Apr 15 2023

Status

approved