A306343 Number T(n,k) of defective (binary) heaps on n elements with k defects; triangle T(n,k), n>=0, 0<=k<=max(0,n-1), read by rows.

1, 1, 1, 1, 2, 2, 2, 3, 9, 9, 3, 8, 28, 48, 28, 8, 20, 90, 250, 250, 90, 20, 80, 360, 1200, 1760, 1200, 360, 80, 210, 1526, 5922, 12502, 12502, 5922, 1526, 210, 896, 7616, 34160, 82880, 111776, 82880, 34160, 7616, 896, 3360, 32460, 185460, 576060, 1017060, 1017060, 576060, 185460, 32460, 3360

Offset

0,5

Comments

A defect in a defective heap is a parent-child pair not having the correct order.

T(n,k) is the number of permutations p of [n] having exactly k indices i in {1,...,n} such that p(i) > p(floor(i/2)).

T(n,0) counts perfect (binary) heaps on n elements (A056971).

Links

Alois P. Heinz, Rows n = 0..190, flattened

Eric Weisstein's World of Mathematics, Heap

Wikipedia, Binary heap

Formula

T(n,k) = T(n,n-1-k) for n > 0.

Sum_{k>=0} k * T(n,k) = A001286(n).

Sum_{k>=0} (k+1) * T(n,k) = A001710(n-1) for n > 0.

Sum_{k>=0} (k+2) * T(n,k) = A038720(n) for n > 0.

Sum_{k>=0} (k+3) * T(n,k) = A229039(n) for n > 0.

Sum_{k>=0} (k+4) * T(n,k) = A230056(n) for n > 0.

Example

T(4,0) = 3: 4231, 4312, 4321.
T(4,1) = 9: 2413, 3124, 3214, 3241, 3412, 3421, 4123, 4132, 4213.
T(4,2) = 9: 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2431, 3142.
T(4,3) = 3: 1234, 1243, 1324.
(The examples use max-heaps.)
Triangle T(n,k) begins:
    1;
    1;
    1,    1;
    2,    2,     2;
    3,    9,     9,     3;
    8,   28,    48,    28,      8;
   20,   90,   250,   250,     90,    20;
   80,  360,  1200,  1760,   1200,   360,    80;
  210, 1526,  5922, 12502,  12502,  5922,  1526,  210;
  896, 7616, 34160, 82880, 111776, 82880, 34160, 7616, 896;
  ...

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:
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 = u + o, g, l},
     If[n == 0, 1, g := 2^Floor@Log[2, n]; 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]]];
T[n_] := CoefficientList[b[n, 0], x];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Feb 17 2021, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A056971, A323957, A323958, A323959, A323960, A323961, A323962, A323963, A323964, A323965, A323966.

Row sums give A000142.

T(n,floor(n/2)) gives A306356.

Cf. A001286, A001710, A008292, A038720, A229039, A230056, A264902, A306393.

Sequence in context: A110910 A119532 A010583 * A238188 A360693 A051007

Adjacent sequences: A306340 A306341 A306342 * A306344 A306345 A306346

Keyword

nonn,tabf

Author

Alois P. Heinz, Feb 08 2019

Status

approved