A213086 Square array read by antidiagonals: T(n,m) (n>=1, m>=0) is the number of partitions of mn that are the sum of m not necessarily distinct partitions of n.
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 4, 1, 1, 7, 14, 10, 5, 1, 1, 11, 25, 30, 15, 6, 1, 1, 15, 53, 65, 55, 21, 7, 1, 1, 22, 89, 173, 140, 91, 28, 8, 1, 1, 30, 167, 343, 448, 266, 140, 36, 9, 1, 1, 42, 278, 778, 1022, 994, 462, 204, 45, 10, 1, 1, 56, 480, 1518, 2710, 2562, 1974, 750, 285, 55, 11, 1
Offset: 1
Examples
The array begins: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ... 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, ... 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, ... 1, 11, 53, 173, 448, 994, 1974, 3606, 6171, 10021, ... 1, 15, 89, 343, 1022, 2562, 5670, 11418, 21351, 37609, ... 1, 22, 167, 778, 2710, 7764, 19314, 43164, 88671, 170170, ... ...
Links
- Alois P. Heinz, Antidiagonals n = 1..18, flattened
- N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.
Crossrefs
Programs
-
Maple
with(combinat): g:= proc(n, m) option remember; `if`(m>1, map(x-> map(y-> sort([x[], y[]]), g(n, 1))[], g(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x), {partition(n)[]}), {[]})) end: T:= (n, m)-> nops(g(n, m)): seq(seq(T(d-m, m), m=0..d-1), d=1..12); # Alois P. Heinz, Jul 11 2012 -
Mathematica
T[n_, m_] := Module[{ip, lg, i}, ip = IntegerPartitions[n]; lg = Length[ ip]; i[0]=1; Table[Join[Sequence @@ Table[ip[[i[k]]], {k, 1, m}]] // Sort, Evaluate[Sequence @@ Table[{i[k], i[k-1], lg}, {k, 1, m}]]] // Flatten[#, m-1]& // Union // Length]; T[_, 0] = 1; Table[T[n-m, m], {n, 1, 12}, {m, 0, n - 1}] // Flatten (* Jean-François Alcover, May 25 2016 *)
Formula
Row n is a polynomial in m: see A213074 for the coefficients.
Extensions
More terms and cross-references from Alois P. Heinz, Jul 11 2012