A327583 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), n>=0, min(j:A001787(j)>=n)<=k<=n, read by rows.

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, 1824, 131304, 1689040, 7805080, 16914786, 17346352, 7087261, 2520, 420084, 8279640, 58728660, 194051060, 333027016

Offset

0,3

Comments

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

Links

Alois P. Heinz, Rows n = 0..150, flattened

Example

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;
           ...

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<h(n), 0, add(b(n-i*j, min(n-i*j, i-1),
        k, p+j)*C(C(k, i), j), j=0..n/i)))
    end:
T:= (n, k)-> add(b(n$2, i, 0)*(-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_, 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], {k, h[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2021, after Alois P. Heinz *)

Crossrefs

Main diagonal gives A000670.

Row sums give A321587.

Column sums give A327585.

Cf. A001787, A326914, A327584 (this triangle read by columns).

Sequence in context: A092417 A071543 A242648 * A324159 A220847 A127611

Adjacent sequences: A327580 A327581 A327582 * A327584 A327585 A327586

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 17 2019

Status

approved