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.

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

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-François 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: A363157 A363952 A245151 * A356898 A106844 A128618

Adjacent sequences: A243975 A243976 A243977 * A243979 A243980 A243981

Keyword

nonn,tabl

Author

Joerg Arndt and Alois P. Heinz, Jun 28 2014

Status

approved