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A373272 Triangle read by rows: T(n,k) = sum of all distinct multiplicities in the integer partitions of n with k parts.

%I A373272 #36 Jun 01 2024 21:03:24
%S A373272 1,1,2,1,1,3,1,3,3,4,1,2,6,4,5,1,4,7,6,5,6,1,3,10,11,10,6,7,1,5,11,16,
%T A373272 14,12,7,8,1,4,15,20,22,14,14,8,9,1,6,16,26,28,29,20,16,9,10,1,5,20,
%U A373272 34,41,40,34,23,18,10,11,1,7,22,42,50,54,44,35,26,20,11,12,1,6,26,52,69,75,68,54,44,29,22,12,13
%N A373272 Triangle read by rows: T(n,k) = sum of all distinct multiplicities in the integer partitions of n with k parts.
%H A373272 Alois P. Heinz, <a href="/A373272/b373272.txt">Rows n = 1..200, flattened</a> (first 40 rows from Olivier Gérard)
%e A373272 Array begins:
%e A373272   1;
%e A373272   1, 2;
%e A373272   1, 1,  3;
%e A373272   1, 3,  3,  4;
%e A373272   1, 2,  6,  4,  5;
%e A373272   1, 4,  7,  6,  5,  6;
%e A373272   1, 3, 10, 11, 10,  6,  7;
%e A373272   1, 5, 11, 16, 14, 12,  7,  8;
%e A373272   1, 4, 15, 20, 22, 14, 14,  8, 9;
%e A373272   1, 6, 16, 26, 28, 29, 20, 16, 9, 10;
%e A373272   ...
%e A373272 T(6,3) = 7 because the partitions of 6 into 3 parts are 4+1+1, 3+2+1, 2+2+2,
%e A373272   the multiplicities are (1,2), (1,1,1), (3),
%e A373272   the distinct multiplicities are respectively (1,2), (1), (3),
%e A373272   contributing 3+1+3 = 7.
%t A373272 Flatten[Table[
%t A373272   Plus @@@
%t A373272    Table[Map[Plus @@ Union[Length /@ Split[#]] &,
%t A373272      IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]]
%Y A373272 Columns k=1-2 give: A057427, A028242.
%Y A373272 Main diagonal gives A000027.
%Y A373272 Row sums are A373273.
%Y A373272 T(2n,n) gives A373104.
%K A373272 nonn,tabl
%O A373272 1,3
%A A373272 _Olivier Gérard_, May 29 2024