A326962 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), k>=0, k<=n<=k*2^(k-1), read by columns.

1, 1, 2, 2, 1, 5, 12, 18, 20, 18, 15, 11, 6, 3, 1, 15, 64, 166, 332, 566, 864, 1214, 1596, 1975, 2320, 2600, 2780, 2842, 2780, 2600, 2320, 1979, 1608, 1238, 908, 626, 404, 246, 136, 69, 32, 12, 4, 1, 52, 340, 1315, 3895, 9770, 21848, 44880, 86275, 157140

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, Columns k = 0..10, flattened

Wikipedia, Partition (number theory)

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.

Example

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;
  ...

Maple

C:= binomial:
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), n=k..k*2^(k-1)), k=0..5);

Mathematica

c = Binomial;
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], {n, k, k 2^(k-1)}], {k, 0, 5}] // Flatten (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Crossrefs

Main diagonal gives A000110.

Row sums give A116539.

Column sums give A003465.

Cf. A001787, A255903, A326914 (this triangle read by rows), A327115, A327116, A327117.

Sequence in context: A329429 A326617 A371727 * A350820 A371886 A280245

Adjacent sequences: A326959 A326960 A326961 * A326963 A326964 A326965

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 13 2019

Status

approved