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A243978 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k. 12
1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 6, 0, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 13, 1, 0, 0, 0, 0, 1, 0, 16, 4, 0, 1, 0, 0, 0, 1, 0, 25, 2, 2, 0, 0, 0, 0, 0, 1, 0, 33, 6, 1, 0, 1, 0, 0, 0, 0, 1, 0, 49, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 61, 9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 90, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 156, 9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

T(0,0) = 1 by convention.

Columns k=0-10 give: A000007, A183558, A244515, A244516, A244517, A244518, A245037, A245038, A245039, A245040, A245041.

Row sums are A000041.

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened)

EXAMPLE

Triangle starts:

00:  1;

01:  0,   1;

02:  0,   1,  1;

03:  0,   2,  0, 1;

04:  0,   3,  1, 0, 1;

05:  0,   6,  0, 0, 0, 1;

06:  0,   7,  2, 1, 0, 0, 1;

07:  0,  13,  1, 0, 0, 0, 0, 1;

08:  0,  16,  4, 0, 1, 0, 0, 0, 1;

09:  0,  25,  2, 2, 0, 0, 0, 0, 0, 1;

10:  0,  33,  6, 1, 0, 1, 0, 0, 0, 0, 1;

11:  0,  49,  4, 2, 0, 0, 0, 0, 0, 0, 0, 1;

12:  0,  61,  9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;

13:  0,  90,  6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;

14:  0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;

15:  0, 156,  9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

16:  0, 198, 23, 3, 4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;

17:  0, 269, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

18:  0, 334, 34, 9, 3, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;

19:  0, 448, 27, 8, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

20:  0, 556, 51, 7, 6, 3, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;

...

The A000041(9) = 30 partitions of 9 with the least multiplicities of any part are:

01:  [ 1 1 1 1 1 1 1 1 1 ]   9

02:  [ 1 1 1 1 1 1 1 2 ]   1

03:  [ 1 1 1 1 1 1 3 ]   1

04:  [ 1 1 1 1 1 2 2 ]   2

05:  [ 1 1 1 1 1 4 ]   1

06:  [ 1 1 1 1 2 3 ]   1

07:  [ 1 1 1 1 5 ]   1

08:  [ 1 1 1 2 2 2 ]   3

09:  [ 1 1 1 2 4 ]   1

10:  [ 1 1 1 3 3 ]   2

11:  [ 1 1 1 6 ]   1

12:  [ 1 1 2 2 3 ]   1

13:  [ 1 1 2 5 ]   1

14:  [ 1 1 3 4 ]   1

15:  [ 1 1 7 ]   1

16:  [ 1 2 2 2 2 ]   1

17:  [ 1 2 2 4 ]   1

18:  [ 1 2 3 3 ]   1

19:  [ 1 2 6 ]   1

20:  [ 1 3 5 ]   1

21:  [ 1 4 4 ]   1

22:  [ 1 8 ]   1

23:  [ 2 2 2 3 ]   1

24:  [ 2 2 5 ]   1

25:  [ 2 3 4 ]   1

26:  [ 2 7 ]   1

27:  [ 3 3 3 ]   3

28:  [ 3 6 ]   1

29:  [ 4 5 ]   1

30:  [ 9 ]   1

Therefore row n=9 is [0, 25, 2, 2, 0, 0, 0, 0, 0, 1].

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))

    end:

T:= (n, k)-> b(n$2, k) -`if`(n=0 and k=0, 0, b(n$2, k+1)):

seq(seq(T(n, k), k=0..n), n=0..14);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-i*j, i-1, k], {j, Max[1, k], n/i}]]]; T[n_, k_] := b[n, n, k] - If[n == 0 && k == 0, 0, b[n, n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Jan 08 2015, translated from Maple *)

CROSSREFS

Cf. A183568, A242451 (the same for compositions).

Cf. A091602 (partitions by max multiplicity of any part).

Sequence in context: A286470 A243055 A245151 * A106844 A125989 A125924

Adjacent sequences:  A243975 A243976 A243977 * A243979 A243980 A243981

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Jun 28 2014

STATUS

approved

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Last modified January 22 10:32 EST 2018. Contains 298042 sequences.