A104778 - OEIS

%I #27 Feb 25 2023 08:33:53

%S 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,

%T 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,

%U 24,30,40,56,43,73,103,136,196,382,764,1

%N 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).

%C Row sums give A178718.

%H Wouter Meeussen, <a href="/A104778/b104778.txt">Table of n, a(n) for n = 0..372</a>

%H Wouter Meeussen, <a href="/A104778/a104778.txt">Table of n, a(n), partition parts for n = 0..372</a>

%e The 47 multinomials (corresponding to A005651(4)=47) can be distributed as in the following triangular array:

%e   1

%e   9 1

%e   4 6 1

%e   9 2 3 1

%e   1 3 2 3 1

%e divide each term by

%e   1

%e   3 1

%e   2 3 1

%e   3 2 3 1

%e   1 3 2 3 1

%e yielding

%e   1

%e   3 1

%e   2 2 1

%e   3 1 1 1

%e   1 1 1 1 1

%e with column sums 10 5 3 2 1.

%e Therefore the fourth row of the table is 1 2 3 5 10

%e The initial rows are:

%e   1,

%e   1,

%e   1, 2,

%e   1, 2, 4,

%e   1, 2, 3, 5, 10,

%e   1, 2, 3, 5, 7, 13, 26,

%e   1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76,

%e   1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232,

%e   1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764,

%e   ...

%t (* for function 'kostka' see A178718 *)

%t aspartitions[n_] := Reverse /@ Sort[Sort /@ Partitions[n]];

%t asorder[n_] := rankpartition /@ Reverse /@ Sort[Sort /@ Partitions[n]];

%t Flatten[Table[Tr/@ Transpose[PadLeft[#,PartitionsP[k]] [[asorder[k]] ]&/@ kostka/@ aspartitions[k]],{k,11}]]

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

%Y A001475 and A000085 are subsequences.

%K nonn,tabf

%O 0,4

%A _Alford Arnold_, Mar 24 2005

%E Corrected and edited by _Wouter Meeussen_, Jan 15 2012