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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).

%I A104778 #27 Feb 25 2023 08:33:53
%S A104778 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 A104778 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 A104778 24,30,40,56,43,73,103,136,196,382,764,1
%N 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).
%C A104778 Row sums give A178718.
%H A104778 Wouter Meeussen, <a href="/A104778/b104778.txt">Table of n, a(n) for n = 0..372</a>
%H A104778 Wouter Meeussen, <a href="/A104778/a104778.txt">Table of n, a(n), partition parts for n = 0..372</a>
%e A104778 The 47 multinomials (corresponding to A005651(4)=47) can be distributed as in the following triangular array:
%e A104778   1
%e A104778   9 1
%e A104778   4 6 1
%e A104778   9 2 3 1
%e A104778   1 3 2 3 1
%e A104778 divide each term by
%e A104778   1
%e A104778   3 1
%e A104778   2 3 1
%e A104778   3 2 3 1
%e A104778   1 3 2 3 1
%e A104778 yielding
%e A104778   1
%e A104778   3 1
%e A104778   2 2 1
%e A104778   3 1 1 1
%e A104778   1 1 1 1 1
%e A104778 with column sums 10 5 3 2 1.
%e A104778 Therefore the fourth row of the table is 1 2 3 5 10
%e A104778 The initial rows are:
%e A104778   1,
%e A104778   1,
%e A104778   1, 2,
%e A104778   1, 2, 4,
%e A104778   1, 2, 3, 5, 10,
%e A104778   1, 2, 3, 5, 7, 13, 26,
%e A104778   1, 2, 3, 4, 5, 8, 11, 14, 20, 38, 76,
%e A104778   1, 2, 3, 4, 5, 8, 10, 13, 14, 23, 32, 42, 60, 116, 232,
%e A104778   1, 2, 3, 4, 5, 5, 8, 11, 14, 17, 14, 24, 30, 40, 56, 43, 73, 103, 136, 196, 382, 764,
%e A104778   ...
%t A104778 (* for function 'kostka' see A178718 *)
%t A104778 aspartitions[n_] := Reverse /@ Sort[Sort /@ Partitions[n]];
%t A104778 asorder[n_] := rankpartition /@ Reverse /@ Sort[Sort /@ Partitions[n]];
%t A104778 Flatten[Table[Tr/@ Transpose[PadLeft[#,PartitionsP[k]] [[asorder[k]] ]&/@ kostka/@ aspartitions[k]],{k,11}]]
%Y A104778 Cf. A000041, A000085, A005651, A036038, A097522, A104707, A104778, A178718.
%Y A104778 A001475 and A000085 are subsequences.
%K A104778 nonn,tabf
%O A104778 0,4
%A A104778 _Alford Arnold_, Mar 24 2005
%E A104778 Corrected and edited by _Wouter Meeussen_, Jan 15 2012