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A306356
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Number of defective (binary) heaps on n elements with floor(n/2) defects.
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2
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1, 1, 1, 2, 9, 48, 250, 1760, 12502, 111776, 1017060, 11165280, 123760560, 1602344832, 21025461600, 314958758400, 4765553385120, 80958196300800, 1386261729792960, 26344715667079680, 502986050203680000, 10556482426015426560, 222685725334400064000
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OFFSET
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0,4
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COMMENTS
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Or number of permutations p of [n] having exactly floor(n/2) indices i in {1,...,n} such that p(i) > p(floor(i/2)).
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LINKS
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Eric Weisstein's World of Mathematics, Heap
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FORMULA
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EXAMPLE
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a(2) = 1: 12.
a(3) = 2: 213, 231.
a(4) = 9: 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2431, 3142.
a(5) = 48: 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 24135, 24153, 24513, 24531, 25314, 25341, 25413, 25431, 31245, 31254, 32145, 32154, 32415, 32451, 32514, 32541, 34125, 34152, 34215, 34251, 34512, 34521, 35412, 35421, 41235, 41253, 41325, 41352, 42135, 42153, 42513, 42531, 51234, 51243, 51324, 51342, 51423, 51432, 52134, 52143.
(The examples use max-heaps.)
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MAPLE
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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, iquo(n, 2)):
seq(a(n), n=0..25);
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MATHEMATICA
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b[u_, o_] := b[u, o] = Module[{n, g, l}, n = u+o; 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, Quotient[n, 2]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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