A273693 Number A(n,k) of k-ary heaps on n elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 2, 3, 1, 0, 1, 1, 1, 2, 6, 8, 1, 0, 1, 1, 1, 2, 6, 12, 20, 1, 0, 1, 1, 1, 2, 6, 24, 40, 80, 1, 0, 1, 1, 1, 2, 6, 24, 60, 180, 210, 1, 0, 1, 1, 1, 2, 6, 24, 120, 240, 630, 896, 1, 0, 1, 1, 1, 2, 6, 24, 120, 360, 1260, 3360, 3360, 1, 0

Offset

0,19

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Wikipedia, D-ary heap

Example

A(4,2) = 3: 1234, 1243, 1324.
A(5,2) = 8: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254.
A(5,3) = 12: 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14325.
A(6,3) = 40: 123456, 123465, 123546, 123564, 123645, 123654, 124356, 124365, 124536, 124563, 124635, 124653, 125346, 125364, 125436, 125463, 125634, 125643, 126345, 126354, 126435, 126453, 126534, 126543, 132456, 132465, 132546, 132564, 132645, 132654, 134256, 134265, 135246, 135264, 136245, 136254, 142356, 142365, 143256, 143265.
(The examples use min-heaps.)
Square array A(n,k) begins:
  1, 1,   1,   1,    1,    1,    1,    1, ...
  1, 1,   1,   1,    1,    1,    1,    1, ...
  0, 1,   1,   1,    1,    1,    1,    1, ...
  0, 1,   2,   2,    2,    2,    2,    2, ...
  0, 1,   3,   6,    6,    6,    6,    6, ...
  0, 1,   8,  12,   24,   24,   24,   24, ...
  0, 1,  20,  40,   60,  120,  120,  120, ...
  0, 1,  80, 180,  240,  360,  720,  720, ...
  0, 1, 210, 630, 1260, 1680, 2520, 5040, ...

Maple

with(combinat):
A:= proc(n, k) option remember; local h, i, x, y, z;
      if n<2 then 1 elif k<2 then k
    else h:= ilog[k]((k-1)*n+1);
         if k^h=(k-1)*n+1 then A((n-1)/k, k)^k*
            multinomial(n-1, ((n-1)/k)$k)
       else x, y:=(k^h-1)/(k-1), (k^(h-1)-1)/(k-1);
            for i from 0 do z:= (n-1)-(k-1-i)*y-i*x;
              if y<=z and z<=x then A(y, k)^(k-1-i)*
                 multinomial(n-1, y$(k-1-i), x$i, z)*
                 A(x, k)^i*A(z, k); break fi
            od
      fi fi
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

Mathematica

multinomial[n_, k_] := n!/Times @@ (k!); A[n_, k_] := A[n, k] = Module[{h, i, x, y, z}, Which[n<2, 1, k<2, k, True, h = Floor @ Log[k, (k-1)*n+1]; If[k^h == (k-1)*n+1, A[(n-1)/k, k]^k*multinomial[n-1, Array[((n-1)/k)&, k]], {x, y} = {(k^h-1)/(k-1), (k^(h-1)-1)/(k-1)}; For[i = 0, True, i++, z = (n-1)-(k-1-i)*y-i*x; If[y<=z && z<=x, A[y, k]^(k-1-i)*multinomial[n-1, Join[Array[y&, k-1-i], Array[x&, i], {z}]]*A[x, k]^i*A[z, k] // Return]] ]]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 13 2017, translated from Maple *)

Crossrefs

Columns k=0-10 give: A019590(n+1), A000012, A056971, A178008, A178009, A178010, A178011, A273694, A273695, A273696, A273697.

Main diagonal gives: A000142(n-1) for n>0.

Cf. A273712.

Sequence in context: A353048 A006842 A299038 * A219967 A060505 A336727

Adjacent sequences: A273690 A273691 A273692 * A273694 A273695 A273696

Keyword

nonn,tabl

Author

Alois P. Heinz, May 28 2016

Status

approved