A104778 Table of values with shape sequence A000041 related to involutions and multinomials. Also column sums of the Kostka matrices associated with the partitions (in Abramowitz & Stegun ordering).

1, 1, 1, 2, 1, 2, 4, 1, 2, 3, 5, 10, 1, 2, 3, 5, 7, 13, 26, 1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76, 1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232, 1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764, 1

Offset: 0

Author

Alford Arnold, Mar 24 2005

Keywords

• nonn
• tabf

Comments

Row sums give A178718.

Examples

			The 47 multinomials (corresponding to A005651(4)=47) can be distributed as in the following triangular array:
  1
  9 1
  4 6 1
  9 2 3 1
  1 3 2 3 1
divide each term by
  1
  3 1
  2 3 1
  3 2 3 1
  1 3 2 3 1
yielding
  1
  3 1
  2 2 1
  3 1 1 1
  1 1 1 1 1
with column sums 10 5 3 2 1.
Therefore the fourth row of the table is 1 2 3 5 10
The initial rows are:
  1,
  1,
  1, 2,
  1, 2, 4,
  1, 2, 3, 5, 10,
  1, 2, 3, 5, 7, 13, 26,
  1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76,
  1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232,
  1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764,
  ...
		

Links

• Wouter Meeussen, Table of n, a(n) for n = 0..372
• Wouter Meeussen, Table of n, a(n), partition parts for n = 0..372

Crossrefs

Cf. A000041, A000085, A005651, A036038, A097522, A104707, A104778, A178718.

A001475 and A000085 are subsequences.

Programs

Mathematica

(* for function 'kostka' see A178718 *)
aspartitions[n_] := Reverse /@ Sort[Sort /@ Partitions[n]];
asorder[n_] := rankpartition /@ Reverse /@ Sort[Sort /@ Partitions[n]];
Flatten[Table[Tr/@ Transpose[PadLeft[#,PartitionsP[k]] [[asorder[k]] ]&/@ kostka/@ aspartitions[k]],{k,11}]]

Extensions

Corrected and edited by Wouter Meeussen, Jan 15 2012