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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).
6
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.
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
KEYWORD
nonn,tabf
AUTHOR
Alford Arnold, Mar 24 2005
EXTENSIONS
Corrected and edited by Wouter Meeussen, Jan 15 2012
STATUS
approved