A279038 Triangle of multinomial coefficients read by rows (ordered by decreasing size of the greatest part).

1, 1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 90, 40, 120, 40, 15, 45, 15, 1, 720, 840, 504, 504, 420, 630, 210, 280, 210, 420, 70, 105, 105, 21, 1, 5040, 5760, 3360, 3360, 2688, 4032, 1344, 1260, 3360, 1260, 2520, 420, 1120, 1120, 1680, 1120, 112, 105, 420, 210, 28, 1

Offset: 0

Author

David W. Wilson and Olivier Gérard, Dec 04 2016

Keywords

• nonn
• tabf
• look
• easy

Comments

The ordering of integer partitions used in this version is also called:

- canonical ordering

- graded reverse lexicographic ordering

- magma (software) ordering

by opposition to the ordering used by Abramowitz and Stegun.

Examples

			First rows are:
    1
    1
    1   1
    2   3   1
    6   8   3   6   1
   24  30  20  20  15   10   1
  120 144  90  90  40  120  40  15  45  15  1
  720 840 504 504 420  630 210 280 210 420 70 105 105 21 1
  ...
		

Links

• Alois P. Heinz, Rows n = 0..28, flattened

Crossrefs

Cf. A000041 (number of partitions of n, length of each row).

Cf. A128628 (triangle of partition lengths)

Cf. A036039 (a different ordering), A102189 (row reversed version of A036039).

Row sums give A000142.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1],
      `if`(i<1, [], [seq(map(x-> x*i^j*j!,
       b(n-i*j, i-1))[], j=[iquo(n, i)-t$t=0..n/i])]))
    end:
T:= n-> map(x-> n!/x, b(n$2))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 04 2016

Mathematica

Flatten[Table[
  Map[n!/Times @@ ((First[#]^Length[#]*Factorial[Length[#]]) & /@
        Split[#]) &, IntegerPartitions[n]], {n, 1, 8}]]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1},
     If[i < 1, {}, Flatten@Table[#*i^j*j!& /@
     b[n - i*j, i - 1], {j, Quotient[n, i] - Range[0, n/i]}]]];
T[n_] := n!/#& /@ b[n, n];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Jun 01 2021, after Alois P. Heinz *)

Extensions

One term for row n=0 prepended by Alois P. Heinz, Dec 04 2016