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A323957
Number of defective (binary) heaps on n elements with exactly one defect.
4
0, 1, 2, 9, 28, 90, 360, 1526, 7616, 32460, 190800, 947760, 6382464, 37065600, 296524800, 1812861600, 15283107840, 105015593280, 1017540576000, 7304720544000, 74472335308800, 629300251008000, 7429184791142400, 62417372203929600, 746041213793075200
OFFSET
1,3
COMMENTS
Or number of permutations p of [n] having exactly one index i in {1,...,n} such that p(i) > p(floor(i/2)).
LINKS
Eric Weisstein's World of Mathematics, Heap
Wikipedia, Binary heap
EXAMPLE
a(2) = 1: 12.
a(3) = 2: 213, 231.
a(4) = 9: 2413, 3124, 3214, 3241, 3412, 3421, 4123, 4132, 4213.
a(5) = 28: 25134, 25143, 35124, 35142, 35214, 35241, 42315, 42351, 43125, 43152, 43215, 43251, 43512, 43521, 45123, 45132, 45213, 45231, 45312, 45321, 52314, 52341, 52413, 52431, 53124, 53142, 53214, 53241.
a(6) = 90: 362451, 362541, 436125, 436215, ..., 652314, 652413, 653124, 653214.
(The examples use max-heaps.)
MAPLE
b:= proc(u, o) option remember; local n, g, l; n:= u+o;
if n=0 then 1
else g:= 2^ilog2(n); l:= min(g-1, n-g/2); expand(
add(add(binomial(j-1, i)*binomial(n-j, l-i)*
b(i, l-i)*b(j-1-i, n-l-j+i), i=0..min(j-1, l)), j=1..u)+
add(add(binomial(j-1, i)*binomial(n-j, l-i)*
b(l-i, i)*b(n-l-j+i, j-1-i), i=0..min(j-1, l)), j=1..o)*x)
fi
end:
a:= n-> coeff(b(n, 0), x, 1):
seq(a(n), n=1..25);
MATHEMATICA
b[u_, o_] := b[u, o] = Module[{n = u+o, g, l}, If[n == 0, 1,
g = 2^(Length[IntegerDigits[n, 2]] - 1);
l = Min[g - 1, n - g/2]; Expand[
Sum[ Sum[Binomial[j - 1, i]*Binomial[n - j, l - i]*
b[i, l-i]*b[j-1-i, n-l-j+i], {i, 0, Min[j-1, l]}], {j, 1, u}] +
Sum[Sum[Binomial[j - 1, i]*Binomial[n - j, l - i]*
b[l-i, i]*b[n-l-j+i, j-1-i], {i, 0, Min[j-1, l]}], {j, 1, o}]*x]]];
a[n_] := Coefficient[b[n, 0], x, 1];
Array[a, 25] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)
CROSSREFS
Column k=1 of A306343.
Sequence in context: A368217 A360479 A258347 * A026087 A109188 A248437
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 09 2019
STATUS
approved