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A243978
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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.
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19
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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
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OFFSET
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0,8
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COMMENTS
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T(0,0) = 1 by convention.
Columns k=0-10 give: A000007, A183558, A244515, A244516, A244517, A244518, A245037, A245038, A245039, A245040, A245041.
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LINKS
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EXAMPLE
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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].
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A091602 (partitions by max multiplicity of any part).
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KEYWORD
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AUTHOR
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STATUS
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approved
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