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

0,4

Comments

Row sums give A178718.

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

Example

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,
  ...

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}]]

Crossrefs

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

A001475 and A000085 are subsequences.

Sequence in context: A132082 A129644 A081517 * A356184 A292477 A081532

Adjacent sequences: A104775 A104776 A104777 * A104779 A104780 A104781

Keyword

nonn,tabf

Author

Alford Arnold, Mar 24 2005

Extensions

Corrected and edited by Wouter Meeussen, Jan 15 2012

Status

approved