A213074 - OEIS

%I #45 May 26 2016 02:14:08

%S 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,

%T 49,340,701,431,1,70,685,2190,2756,1188,1,93,1075,3935,5564,2658

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

%C 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.

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

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

%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.

%e Triangle c(n,k) begins:

%e n\k

%e -  0    1    2    3    4    5 ...

%e ---------------------------------

%e 2  1

%e 3  1

%e 4  1    2

%e 5  1    3

%e 6  1    7    8

%e 7  1   10   14

%e 8  1   17   50   36

%e 9  1   24   89   78

%e 10 1   36  207  368  200

%e 11 1   49  340  701  431

%e 12 1   70  685 2190 2756 1188

%e 13 1   93 1075 3935 5564 2658

%e ...

%p with(combinat):

%p h:= proc(n, m) option remember;

%p       `if`(m>1, map(x-> map(y-> sort([x[], y[]]), h(n, 1))[],

%p        h(n, m-1)), `if`(m=1, map(x->map(y-> `if`(y>1, y-1, NULL), x),

%p        {partition(n)[]}), {[]}))

%p     end:

%p T:= proc(n) local i, g, t;

%p       g:= floor((n+1)/2);

%p       subs(solve({seq(nops(h(n, t))=add(c||i *binomial(t+g, g+i),

%p       i=0..n-g-1), t=1..n-g)}, {seq(c||i, i=0..n-g-1)}),

%p       [seq(c||i, i=0..n-g-1)])[]

%p     end:

%p seq(T(n), n=2..10);  # _Alois P. Heinz_, Jul 11 2012

%t nmax = 13; mmax = 5;

%t 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;

%t 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}]]];

%t 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}];

%t cc (* _Jean-François Alcover_, May 25 2016 *)

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

%K nonn,tabf,more

%O 2,4

%A _N. J. A. Sloane_, Jun 04 2012

%E 12 more terms (rows 12-13) from _Alois P. Heinz_, Jul 11 2012