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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

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Multinomial coefficients

Wikipedia, Partition (number theory)

Wikipedia, Partition of a set

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.

Cf. A327801, A327884.

Sequence in context: A129624 A177038 A019975 * A196274 A073871 A120927

Adjacent sequences:  A327866 A327867 A327868 * A327870 A327871 A327872

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 28 2019

STATUS

approved

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Last modified May 14 16:48 EDT 2021. Contains 343898 sequences. (Running on oeis4.)