A327584 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.

1, 1, 3, 4, 6, 13, 48, 150, 300, 666, 936, 1824, 2520, 2160, 5040, 75, 536, 2820, 11144, 41346, 131304, 420084, 1191552, 3427008, 9207456, 23466720, 61522560, 141553560, 345346560, 777152160, 1635096960, 3700806480, 6998261760, 14211912960, 27442437120

Offset: 0

Author

Alois P. Heinz, Sep 17 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) = 4: 2ab1a, 2ab1b, 1a2ab, 1b2ab.
T(3,3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
T(4,2) = 6: 2ab1a1b, 1a2ab1b, 1a1b2ab, 2ab1b1a, 1b2ab1a, 1b1a2ab.
Triangle T(n,k) begins:
  1;
     1;
        3;
        4,  13;
        6,  48,    75;
           150,   536,    541;
           300,  2820,   6320,   4683;
           666, 11144,  50150,  81012,   47293;
           936, 41346, 308080, 903210, 1134952, 545835;
           ...
		

Links

• Alois P. Heinz, Columns k = 0..7, flattened

Crossrefs

Main diagonal gives A000670.

Row sums give A321587.

Column sums give A327585.

Cf. A001787, A326962, A327583 (this triangle read by rows).

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, p) option remember; `if`(n=0, p!,
      `if`(i<1 or k add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);

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_, p_] := b[n, i, k, p] = If[n == 0, p!,
     If[i < 1 || k < h[n], 0, Sum[b[n - i*j, Min[n - i*j, i - 1],
     k, p + j]*c[c[k, i], j], {j, 0, n/i}]]];
T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k - i)*c[k, i], {i, 0, k}];
Table[Table[T[n, k], {n, k, k*2^(k - 1)}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Feb 22 2021, after Alois P. Heinz *)