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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. 19
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.
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
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Jun 28 2014
STATUS
approved

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Last modified February 22 22:30 EST 2024. Contains 370265 sequences. (Running on oeis4.)