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

%I

%S 1,1,1,2,9,48,250,1760,12502,111776,1017060,11165280,123760560,

%T 1602344832,21025461600,314958758400,4765553385120,80958196300800,

%U 1386261729792960,26344715667079680,502986050203680000,10556482426015426560,222685725334400064000

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

%C 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)).

%H Alois P. Heinz, <a href="/A306356/b306356.txt">Table of n, a(n) for n = 0..190</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Heap.html">Heap</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Binary_heap">Binary heap</a>

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

%e a(2) = 1: 12.

%e a(3) = 2: 213, 231.

%e a(4) = 9: 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2431, 3142.

%e 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.

%e (The examples use max-heaps.)

%p b:= proc(u, o) option remember; local n, g, l; n:= u+o;

%p if n=0 then 1

%p else g:= 2^ilog2(n); l:= min(g-1, n-g/2); expand(

%p add(add(binomial(j-1, i)*binomial(n-j, l-i)*

%p b(i, l-i)*b(j-1-i, n-l-j+i), i=0..min(j-1, l)), j=1..u)+

%p add(add(binomial(j-1, i)*binomial(n-j, l-i)*

%p b(l-i, i)*b(n-l-j+i, j-1-i), i=0..min(j-1, l)), j=1..o)*x)

%p fi

%p end:

%p a:= n-> coeff(b(n, 0), x, iquo(n, 2)):

%p seq(a(n), n=0..25);

%Y Cf. A056971, A306343.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Feb 09 2019

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Last modified June 4 11:44 EDT 2020. Contains 334825 sequences. (Running on oeis4.)