A326617 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=A024916(k), read by columns.

1, 1, 2, 2, 1, 5, 9, 9, 10, 9, 3, 13, 44, 96, 152, 155, 124, 140, 160, 113, 48, 16, 4, 42, 225, 680, 1350, 2180, 3751, 6050, 7420, 6870, 5555, 5330, 6300, 6475, 5025, 3000, 1250, 250, 150, 1098, 4155, 11730, 30300, 69042, 127364, 188568, 249690, 365160, 584733

Offset: 0

Author

Alois P. Heinz, Sep 12 2019

Keywords

• nonn
• tabf

Comments

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

Examples

			T(3,2) = 2: 2a1b, 2b1a.
T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1,  9,  13;
            9,  44,   42;
           10,  96,  225,   150;
            9, 152,  680,  1098,    576;
            3, 155, 1350,  4155,   5201,   2266;
               124, 2180, 11730,  26642,  26904,   9966;
               140, 3751, 30300, 106281, 182000, 149832, 47466;
               ...
		

Links

• Alois P. Heinz, Columns k = 0..40, flattened
• Wikipedia, Partition (number theory)

Crossrefs

Main diagonal gives A178682.

Row sums give A326648.

Column sums give A326650.

Cf. A000203, A024916, A326616 (this triangle read by rows), A326649, A326651.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
      b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), n=k..g(k)), k=0..6);

Mathematica

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]] ;
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[Table[T[n, k], {n, k, g[k]}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)

Formula

Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).

Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).