|
%I #34 Jan 18 2022 15:05:17
%S 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,
%T 53,65,55,21,7,1,1,22,89,173,140,91,28,8,1,1,30,167,343,448,266,140,
%U 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
%N 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.
%H Alois P. Heinz, <a href="/A213086/b213086.txt">Antidiagonals n = 1..18, flattened</a>
%H N. Metropolis and P. R. Stein, <a href="http://dx.doi.org/10.1016/S0021-9800(70)80091-6">An elementary solution to a problem in restricted partitions</a>, J. Combin. Theory, 9 (1970), 365-376.
%F Row n is a polynomial in m: see A213074 for the coefficients.
%e The array begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...
%e 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
%e 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, ...
%e 1, 7, 25, 65, 140, 266, 462, 750, 1155, 1705, ...
%e 1, 11, 53, 173, 448, 994, 1974, 3606, 6171, 10021, ...
%e 1, 15, 89, 343, 1022, 2562, 5670, 11418, 21351, 37609, ...
%e 1, 22, 167, 778, 2710, 7764, 19314, 43164, 88671, 170170, ...
%e ...
%p with(combinat):
%p g:= proc(n, m) option remember;
%p `if`(m>1, map(x-> map(y-> sort([x[], y[]]), g(n, 1))[],
%p g(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
%p {partition(n)[]}), {[]}))
%p end:
%p T:= (n, m)-> nops(g(n, m)):
%p seq(seq(T(d-m, m), m=0..d-1), d=1..12); # _Alois P. Heinz_, Jul 11 2012
%t 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;
%t Table[T[n-m, m], {n, 1, 12}, {m, 0, n - 1}] // Flatten (* _Jean-François Alcover_, May 25 2016 *)
%Y Columns are A000041, A002219, A002220, A002221, A002222. Cf. A213074.
%Y Rows are A000027, A000217, A000330, A001296, A207361.
%Y Main diagonal gives A284645.
%K nonn,tabl
%O 1,5
%A _N. J. A. Sloane_, Jun 05 2012
%E More terms and cross-references from _Alois P. Heinz_, Jul 11 2012
|
