A373451 Number T(n,k) of (binary) heaps of length n whose element set equals [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 7, 3, 0, 1, 9, 23, 23, 8, 0, 1, 14, 55, 92, 70, 20, 0, 1, 24, 147, 386, 502, 320, 80, 0, 1, 34, 281, 1004, 1861, 1880, 985, 210, 0, 1, 54, 633, 3128, 8113, 12008, 10237, 4690, 896, 0, 1, 79, 1241, 8039, 27456, 54900, 66730, 48650, 19600, 3360

Offset: 0

Author

Alois P. Heinz, Jun 05 2024

Keywords

• nonn
• tabl

Comments

These heaps may contain repeated elements. Their element sets are gap-free and contain 1 (if nonempty).

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

Examples

			T(4,1) = 1: 1111.
T(4,2) = 5: 2111, 2121, 2211, 2212, 2221.
T(4,3) = 7: 3121, 3211, 3212, 3221, 3231, 3312, 3321.
T(4,4) = 3: 4231, 4312, 4321.
(The examples use max-heaps.)
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   2;
  0, 1,  5,   7,    3;
  0, 1,  9,  23,   23,    8;
  0, 1, 14,  55,   92,   70,    20;
  0, 1, 24, 147,  386,  502,   320,    80;
  0, 1, 34, 281, 1004, 1861,  1880,   985,  210;
  0, 1, 54, 633, 3128, 8113, 12008, 10237, 4690, 896;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened
• Eric Weisstein's World of Mathematics, Heap
• Wikipedia, Binary heap

Crossrefs

Columns k=0-4 give A000007, A057427, A091980(n+1)-2, A376962, A376963.

Row sums give A373452.

Row maxima give A373608.

Main diagonal gives A056971.

First lower diagonal gives A373496.

T(2n,n) gives A373500.

Antidiagonal sums give A373632.

Antidiagonal maxima give A373897.

Cf. A019590, A373449.

Programs

Maple

b:= proc(n, k) option remember; `if`(n=0, 1,
     (g-> (f-> add(b(f, j)*b(n-1-f, j), j=1..k)
             )(min(g-1, n-g/2)))(2^ilog2(n)))
    end:
T:= (n, k)-> add(binomial(k, j)*(-1)^j*b(n, k-j), j=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1,
   Function[g, Function[f, Sum[b[f, j]*b[n-1-f, j], {j, 1, k}]][
      Min[g-1, n-g/2]]][2^(Length@IntegerDigits[n, 2]-1)]];
T[n_, k_] := Sum[Binomial[k, j]*(-1)^j*b[n, k-j], {j, 0, k}];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 20 2024, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{j=0..k} binomial(k,j) * (-1)^j * A373449(n,k-j).

Sum_{k=0..n} (-1)^k * T(n,n-k) = A019590(n+1).