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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. 9
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 (list; graph; refs; listen; history; text; internal format)
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..200, 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:

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-Fran├žois Alcover, Dec 17 2020, after Alois P. Heinz *)

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.

Sequence in context: A326616 A249033 A068762 * A155679 A319771 A021448

Adjacent sequences:  A326911 A326912 A326913 * A326915 A326916 A326917

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Sep 13 2019

STATUS

approved

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Last modified January 16 15:53 EST 2021. Contains 340206 sequences. (Running on oeis4.)