A273730 Square array read by antidiagonals: A(n,k) = number of permutations of n elements divided by the number of k-ary heaps on n+1 elements, n>=0, k>=1.

1, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 24, 1, 1, 1, 1, 3, 120, 1, 1, 1, 1, 2, 6, 720, 1, 1, 1, 1, 1, 3, 9, 5040, 1, 1, 1, 1, 1, 2, 4, 24, 40320, 1, 1, 1, 1, 1, 1, 3, 8, 45, 362880, 1, 1, 1, 1, 1, 1, 2, 4, 12, 108, 3628800, 1, 1, 1, 1, 1, 1, 1, 3, 5, 16, 189, 39916800

Offset

0,6

Links

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

Wikipedia, D-ary heap

Formula

A(n,k) = A000142(n)/A273693(n+1,k).

Example

Square array A(n,k) begins:
:     1,  1,  1, 1, 1, 1, 1, 1, ...
:     1,  1,  1, 1, 1, 1, 1, 1, ...
:     2,  1,  1, 1, 1, 1, 1, 1, ...
:     6,  2,  1, 1, 1, 1, 1, 1, ...
:    24,  3,  2, 1, 1, 1, 1, 1, ...
:   120,  6,  3, 2, 1, 1, 1, 1, ...
:   720,  9,  4, 3, 2, 1, 1, 1, ...
:  5040, 24,  8, 4, 3, 2, 1, 1, ...
: 40320, 45, 12, 5, 4, 3, 2, 1, ...

Maple

with(combinat):
b:= 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 b((n-1)/k, k)^k*
            multinomial(n-1, ((n-1)/k)$k)
       else x, y:=(k^h-1)/(k-1), (k^(h-1)-1)/(k-1);
            for i from 0 do z:= (n-1)-(k-1-i)*y-i*x;
              if y<=z and z<=x then b(y, k)^(k-1-i)*
                 multinomial(n-1, y$(k-1-i), x$i, z)*
                 b(x, k)^i*b(z, k); break fi
            od
      fi fi
    end:
A:= (n, k)-> n!/b(n+1, k):
seq(seq(A(n, 1+d-n), n=0..d), d=0..14);

Mathematica

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, k_] := b[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, b[(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, b[y, k]^(k-1-i)*multinomial[n-1, Join[Array[y&, k-1-i], Array[x&, i], {z}]] * b[x, k]^i*b[z, k] // Return]]]]]; A[n_, k_] :=  n!/b[n+1, k]; Table[A[n, 1+d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 13 2017, translated from Maple *)

Crossrefs

Columns k=1-10 give: A000142, A133385, A273731, A273732, A273733, A273734, A273735, A273736, A273737, A273738.

Cf. A273693.

Sequence in context: A351699 A260685 A351429 * A369964 A326047 A280491

Adjacent sequences: A273727 A273728 A273729 * A273731 A273732 A273733

Keyword

nonn,tabl

Author

Alois P. Heinz, May 28 2016

Status

approved