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A271424 Number T(n,k) of set partitions of [n] with minimal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 3, 0, 1, 0, 51, 0, 0, 0, 1, 0, 132, 55, 15, 0, 0, 1, 0, 771, 105, 0, 0, 0, 0, 1, 0, 3089, 945, 0, 105, 0, 0, 0, 1, 0, 18388, 1218, 1540, 0, 0, 0, 0, 0, 1, 0, 96423, 15456, 3150, 0, 945, 0, 0, 0, 0, 1, 0, 627529, 26785, 24255, 0, 0, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,8
COMMENTS
At least one block length occurs exactly k times, and all block lengths occur at least k times.
LINKS
EXAMPLE
T(4,1) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,2) = 3: 12|34, 13|24, 14|23.
T(4,4) = 1: 1|2|3|4.
T(6,3) = 15: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 4, 0, 1;
0, 11, 3, 0, 1;
0, 51, 0, 0, 0, 1;
0, 132, 55, 15, 0, 0, 1;
0, 771, 105, 0, 0, 0, 0, 1;
0, 3089, 945, 0, 105, 0, 0, 0, 1;
0, 18388, 1218, 1540, 0, 0, 0, 0, 0, 1;
0, 96423, 15456, 3150, 0, 945, 0, 0, 0, 0, 1;
MAPLE
with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
*b(n-i*j, i-1, k)/j!, j={0, $k..n/i})))
end:
T:= (n, k)-> b(n$2, k)-`if`(n=k, 0, b(n$2, k+1)):
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]* b[n-i*j, i-1, k]/j!, {j, Join[{0}, Range[k, n/i]] // Union}]]]; T[n_, k_] := b[n, n, k] - If[n == k, 0, b[n, n, k + 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 16 2017, adapted from Maple *)
CROSSREFS
Row sums give A000110.
Main diagonal gives A000012.
T(2n,n) gives A001147.
T(3n,n) gives A271715.
Cf. A271423.
Sequence in context: A363973 A046781 A244530 * A117435 A282252 A268367
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 07 2016
STATUS
approved

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Last modified March 5 07:51 EST 2024. Contains 370538 sequences. (Running on oeis4.)