A213074 Irregular triangle read by rows: coefficients c(n,k) (n>=2, 0<=k<= floor((n-2)/2)) in formula for number of restricted partitions.

1, 1, 1, 2, 1, 3, 1, 7, 8, 1, 10, 14, 1, 17, 50, 36, 1, 24, 89, 78, 1, 36, 207, 368, 200, 1, 49, 340, 701, 431, 1, 70, 685, 2190, 2756, 1188, 1, 93, 1075, 3935, 5564, 2658

Offset: 2

Author

N. J. A. Sloane, Jun 04 2012

Keywords

• nonn
• tabf
• more

Comments

Let T^(n)_m denote the number of partitions of mn that can be obtained by adding together m (not necessarily distinct) partitions of n (see A213086). For T^(n)_2, T^(n)_3, T^(n)_4, T^(n)_5 see A002219 through A002222.

Metropolis and Stein show that T^(n)_m is given by the formula

T^(n)m = Sum{k=0..n-g-1} binomial(m+g,g+k) c(n,k), where g = floor((n+1)/2).

Examples

			Triangle c(n,k) begins:
n\k
-  0    1    2    3    4    5 ...
---------------------------------
2  1
3  1
4  1    2
5  1    3
6  1    7    8
7  1   10   14
8  1   17   50   36
9  1   24   89   78
10 1   36  207  368  200
11 1   49  340  701  431
12 1   70  685 2190 2756 1188
13 1   93 1075 3935 5564 2658
...
		

Links

• N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.

Crossrefs

Cf. A000041, A002219, A002220, A002221, A002222, A213075, A213076, A213086.

Programs

Maple

with(combinat):
h:= proc(n, m) option remember;
      `if`(m>1, map(x-> map(y-> sort([x[], y[]]), h(n, 1))[],
       h(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),
       {partition(n)[]}), {[]}))
    end:
T:= proc(n) local i, g, t;
      g:= floor((n+1)/2);
      subs(solve({seq(nops(h(n, t))=add(c||i *binomial(t+g, g+i),
      i=0..n-g-1), t=1..n-g)}, {seq(c||i, i=0..n-g-1)}),
      [seq(c||i, i=0..n-g-1)])[]
    end:
seq(T(n), n=2..10);  # Alois P. Heinz, Jul 11 2012

Mathematica

nmax = 13; mmax = 5;
T[n_, m_] := 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;
U[n_, m_] := With[{g = Floor[(n + 1)/2]}, If[n == 1, 1, Sum[Binomial[m + g, g + k] c[n, k], {k, 0, n - g - 1}]]];
Do[TT = Table[T[n , m] - U[n , m], {n, 1, nmax}, {m, 0, mm}] // Flatten; c[_, 0] = 1; sol = Solve[Thread[TT == 0]][[1]]; cc = Table[c[n, k], {n, 2, nmax}, {k, 0, Floor[(n - 2)/2]}] /. sol // Flatten; Print[cc], {mm, 2, mmax}];
cc (* Jean-François Alcover, May 25 2016 *)

Extensions

12 more terms (rows 12-13) from Alois P. Heinz, Jul 11 2012