A326914 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that all parts have different color patterns and a pattern for part i has i distinct colors in increasing order; triangle T(n,k), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

1, 1, 2, 2, 5, 1, 12, 15, 18, 64, 52, 20, 166, 340, 203, 18, 332, 1315, 1866, 877, 15, 566, 3895, 9930, 10710, 4140, 11, 864, 9770, 39960, 74438, 64520, 21147, 6, 1214, 21848, 134871, 386589, 564508, 408096, 115975, 3, 1596, 44880, 402756, 1668338, 3652712

Offset: 0

Author

Alois P. Heinz, Sep 13 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(4,3) = 12: 3abc1a, 3abc1b, 3abc1c, 2ab2ac, 2ab2bc, 2ac2bc, 2ab1a1c, 2ab1b1c, 2ac1a1b, 2ac1b1c, 2bc1a1b, 2bc1a1c.
Triangle T(n,k) begins:
  1;
     1;
        2;
        2,  5;
        1, 12,   15;
           18,   64,    52;
           20,  166,   340,    203;
           18,  332,  1315,   1866,    877;
           15,  566,  3895,   9930,  10710,   4140;
           11,  864,  9770,  39960,  74438,  64520,  21147;
            6, 1214, 21848, 134871, 386589, 564508, 408096, 115975;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened
• Wikipedia, Partition (number theory)

Crossrefs

Main diagonal gives A000110.

Row sums give A116539.

Column sums give A003465.

Cf. A001787, A255903, A326962 (this triangle read by columns), A327115, A327116, A327117.

Programs

Maple

C:= binomial:
g:= proc(n) option remember; n*2^(n-1) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), k=h(n)..n), n=0..12);

Mathematica

c = Binomial;
g[n_] := g[n] = n*2^(n - 1);
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++, If[g[k] >= n, Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i - 1], k] c[c[k, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}];
Table[Table[T[n, k], {k, h[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A327115(n).

T(n*2^(n-1),n) = T(A001787(n),n) = 1.

T(n*2^(n-1)-1,n) = n for n >= 2.