A306356 Number of defective (binary) heaps on n elements with floor(n/2) defects.

1, 1, 1, 2, 9, 48, 250, 1760, 12502, 111776, 1017060, 11165280, 123760560, 1602344832, 21025461600, 314958758400, 4765553385120, 80958196300800, 1386261729792960, 26344715667079680, 502986050203680000, 10556482426015426560, 222685725334400064000

Offset: 0

Author

Alois P. Heinz, Feb 09 2019

Keywords

• nonn

Comments

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)).

Examples

			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.)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..190
• Eric Weisstein's World of Mathematics, Heap
• Wikipedia, Binary heap

Crossrefs

Cf. A056971, A306343.

Programs

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, iquo(n, 2)):
seq(a(n), n=0..25);

Mathematica

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]];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 26 2021, after Alois P. Heinz *)

Formula

a(n) = A306343(n,floor(n/2)).