login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A285824 Number T(n,k) of ordered set partitions of [n] into k blocks such that equal-sized blocks are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 14
1, 0, 1, 0, 1, 1, 0, 1, 6, 1, 0, 1, 11, 18, 1, 0, 1, 30, 75, 40, 1, 0, 1, 52, 420, 350, 75, 1, 0, 1, 126, 1218, 3080, 1225, 126, 1, 0, 1, 219, 4242, 17129, 15750, 3486, 196, 1, 0, 1, 510, 14563, 82488, 152355, 63756, 8526, 288, 1, 0, 1, 896, 42930, 464650, 1049895, 954387, 217560, 18600, 405, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

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

Wikipedia, Partition of a set

EXAMPLE

T(3,1) = 1: 123.

T(3,2) = 6: 1|23, 23|1, 2|13, 13|2, 3|12, 12|3.

T(3,3) = 1: 1|2|3.

Triangle T(n,k) begins:

  1;

  0, 1;

  0, 1,   1;

  0, 1,   6,    1;

  0, 1,  11,   18,     1;

  0, 1,  30,   75,    40,     1;

  0, 1,  52,  420,   350,    75,    1;

  0, 1, 126, 1218,  3080,  1225,  126,   1;

  0, 1, 219, 4242, 17129, 15750, 3486, 196, 1;

MAPLE

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,

      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*x^j*combinat

      [multinomial](n, n-i*j, i$j)/j!^2, j=0..n/i)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):

seq(T(n), n=0..12);

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!);

b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n-i*j, i-1, p+j]*x^j*multinomial[n, Join[{n-i*j}, Table[i, j]]]/ j!^2, {j, 0, n/i}]]];

T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Apr 28 2018, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A057427, A285917, A285918, A285919, A285920, A285921, A285922, A285923, A285924, A285925.

Main diagonal and first lower diagonal give: A000012, A002411.

Row sums give A120774.

T(2n,n) gives A285926.

Cf. A048993, A131689, A226874, A285849.

Sequence in context: A196603 A318458 A267479 * A269955 A198754 A243317

Adjacent sequences:  A285821 A285822 A285823 * A285825 A285826 A285827

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 27 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 1 02:38 EDT 2022. Contains 357134 sequences. (Running on oeis4.)