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A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 9
1, 1, 1, 1, 0, 1, 4, 3, 3, 1, 5, 4, 0, 4, 1, 16, 5, 10, 10, 5, 1, 82, 66, 75, 60, 15, 6, 1, 169, 112, 126, 35, 140, 21, 7, 1, 541, 456, 196, 336, 280, 224, 28, 8, 1, 2272, 765, 1548, 1848, 1386, 630, 336, 36, 9, 1, 17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.
Number T(n,k) of set partitions of [n] with distinct block sizes and one of the block sizes is k. T(5,3) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.
LINKS
EXAMPLE
Triangle T(n,k) begins:
1;
1, 1;
1, 0, 1;
4, 3, 3, 1;
5, 4, 0, 4, 1;
16, 5, 10, 10, 5, 1;
82, 66, 75, 60, 15, 6, 1;
169, 112, 126, 35, 140, 21, 7, 1;
541, 456, 196, 336, 280, 224, 28, 8, 1;
2272, 765, 1548, 1848, 1386, 630, 336, 36, 9, 1;
17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1;
...
MAPLE
with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0),
l=select(x-> nops(x)=nops({x[]}) and
(k=0 or k in x), partition(n))):
seq(seq(T(n, k), k=0..n), n=0..11);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, 1, `if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
`if`(i=k, 0, b(n-i, min(n-i, i-1), k)/i!)))
end:
T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..11);
MATHEMATICA
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n==0, 1, If[i<2, 0, b[n, i-1, If[i==k, 0, k]]] + If[i==k, 0, b[n-i, Min[n-i, i-1], k]/i!]]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, from 2nd Maple program *)
CROSSREFS
Columns k=0-3 give: A007837, A327876, A327881, A328155.
Row sums give A327870.
T(2n,n) gives A328156.
Sequence in context: A129624 A177038 A019975 * A196274 A073871 A120927
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 28 2019
STATUS
approved

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Last modified March 29 06:44 EDT 2024. Contains 371265 sequences. (Running on oeis4.)