login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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.
2

%I #12 Oct 26 2023 09:54:41

%S 1,1,1,1,2,2,4,6,9,12,19,28,44,66,110,170,255,396,651,1001,1519,2520,

%T 4032,6216,9752,15912,25236,38760,63090,101850,160050,248710,408760,

%U 653752,1021735,1634776,2656511,4218786,6562556,10737090,17299646,27313650,43249115

%N 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 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 Robert Israel, <a href="/A362260/b362260.txt">Table of n, a(n) for n = 0..4771</a>

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

%e 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 f:= proc(n) local k, v, m,w;

%p m:= 0:

%p for k from 0 to n/2 do

%p v:= binomial(n-k,k);

%p if n:: even and k::even then w:= binomial((n-k)/2,k/2)

%p elif (n-k)::odd then w:=binomial((n-k-1)/2, floor(k/2))

%p else w:= 0

%p fi;

%p m:= max(m,(v+w)/2);

%p od;

%p m

%p end proc:

%p map(f, [$0..50]); # _Robert Israel_, Oct 25 2023

%Y Row maxima of A102541.

%Y Second column of A362258.

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

%K nonn

%O 0,5

%A _Pontus von Brömssen_, Apr 15 2023