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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- Wikipedia, D-ary heap
Crossrefs
Programs
-
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 *)