A350788 Irregular triangle read by rows: T(n,k) is the number of partial functions on [n] such that the sizes of the preimages of the individual elements in the range form the k-th partition in the class of all partitions listed in Abramowitz and Stegun order, n>=0, 0<=k<=A000070(n).

1, 1, 1, 1, 4, 2, 2, 1, 9, 9, 18, 3, 18, 6, 1, 16, 24, 72, 16, 144, 96, 4, 48, 36, 144, 24, 1, 25, 50, 200, 50, 600, 600, 25, 400, 300, 1800, 600, 5, 100, 200, 600, 900, 1200, 120, 1, 36, 90, 450, 120, 1800, 2400, 90, 1800, 1350, 10800, 5400, 36, 900, 1800, 7200, 10800, 21600, 4320, 6, 180, 450, 1800, 300, 7200, 7200, 1800, 16200, 10800, 720

Offset: 0

Author

Geoffrey Critzer, Jan 16 2022

Keywords

• nonn
• tabf

Comments

The last A000041(n) entries of each row give A049009.

Row sums are (n+1)^n = A000169(n+1).

Examples

			  1,
  1,  1,
  1,  4,  2,  2,
  1,  9,  9, 18,  3,  18,  6,
  1, 16, 24, 72, 16, 144, 96, 4, 48, 36, 144, 24
		

Crossrefs

Cf. A000169, A049009, A000041, A000070.

Programs

Mathematica

g[n_, list_] := Multinomial @@ Join[{n - Length[list]}, Table[Count[list, i], {i, 1, n}]]* Multinomial @@ Join[{n - Total[list]}, list]; Table[Map[g[nn, #] &,
   Level[Table[IntegerPartitions[k], {k, 0, nn}], {2}]], {nn, 0, 5}] // Grid