OFFSET
0,5
COMMENTS
T(n,k) is the number of permutations p of [n] having exactly k pairs (i,j) in {1,...,n} X {1,...,floor(log_2(i))} such that p(i) > p(floor(i/2^j)).
T(n,0) counts perfect (binary) heaps on n elements (A056971).
LINKS
Alois P. Heinz, Rows n = 0..100, flattened
Marko Riedel, math.stackexchange.com, Average number of inversions in a random binary heap on 2^n-1 elements.
Eric Weisstein's World of Mathematics, Heap,
Wikipedia, Binary heap.
Wikipedia, Permutation.
EXAMPLE
T(4,0) = 3: 4231, 4312, 4321.
T(4,1) = 6: 3241, 3412, 3421, 4123, 4132, 4213.
T(4,2) = 6: 2341, 2413, 2431, 3124, 3142, 3214.
T(4,3) = 6: 1342, 1423, 1432, 2134, 2143, 2314.
T(4,4) = 3: 1234, 1243, 1324.
T(5,1) = 16: 43512, 43521, 45123, 45132, 45213, 45231, 45312, 45321, 52314, 52341, 52413, 52431, 53124, 53142, 53214, 53241.
(The examples use max-heaps.)
Triangle T(n,k) begins:
1;
1;
1, 1;
2, 2, 2;
3, 6, 6, 6, 3;
8, 16, 24, 24, 24, 16, 8;
20, 60, 100, 120, 120, 120, 100, 60, 20;
80, 240, 480, 640, 720, 720, 720, 640, 480, 240, 80;
...
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(x^(n-j)*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(x^(j-1)*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))
fi
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);
MATHEMATICA
b[u_, o_] := b[u, o] = Module[{n, g, l}, n = u + o;
If[n == 0, 1, g = 2^Floor@Log[2, n]; l = Min[g - 1, n - g/2]; Expand[
Sum[x^(n-j)*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[x^(j-1)*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}]]]];
T[n_] := CoefficientList[b[n, 0], x];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Feb 15 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Feb 12 2019
STATUS
approved