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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.

%I A362260 #12 Oct 26 2023 09:54:41
%S A362260 1,1,1,1,2,2,4,6,9,12,19,28,44,66,110,170,255,396,651,1001,1519,2520,
%T A362260 4032,6216,9752,15912,25236,38760,63090,101850,160050,248710,408760,
%U A362260 653752,1021735,1634776,2656511,4218786,6562556,10737090,17299646,27313650,43249115
%N 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.
%C A362260 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.
%H A362260 Robert Israel, <a href="/A362260/b362260.txt">Table of n, a(n) for n = 0..4771</a>
%F A362260 a(n) >= A073028(n)/2.
%e A362260 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.
%p A362260 f:= proc(n) local k, v, m,w;
%p A362260   m:= 0:
%p A362260   for k from 0 to n/2 do
%p A362260     v:= binomial(n-k,k);
%p A362260     if n:: even and k::even then w:= binomial((n-k)/2,k/2)
%p A362260     elif (n-k)::odd then w:=binomial((n-k-1)/2, floor(k/2))
%p A362260     else w:= 0
%p A362260     fi;
%p A362260     m:= max(m,(v+w)/2);
%p A362260   od;
%p A362260   m
%p A362260 end proc:
%p A362260 map(f, [$0..50]); # _Robert Israel_, Oct 25 2023
%Y A362260 Row maxima of A102541.
%Y A362260 Second column of A362258.
%Y A362260 Cf. A001224, A073028, A361224 (rectangular pieces).
%K A362260 nonn
%O A362260 0,5
%A A362260 _Pontus von Brömssen_, Apr 15 2023