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Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.
3

%I #23 Jun 02 2024 19:28:56

%S 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,4,3,2,1,1,3,7,6,4,2,1,1,4,8,8,6,4,

%T 2,1,1,4,10,12,10,5,4,2,1,1,5,12,15,13,11,6,4,2,1,1,5,15,21,20,17,11,

%U 6,4,2,1,1,6,16,25,26,21,16,10,6,4,2,1,1,6,20,33,36,34,24,17,11,6,4,2,1,1,7,22,38,46,44,34,25,17,11,6,4,2,1,1,7,25,48,58,56,50,38,24,16,11,6,4,2,1

%N Triangle read by rows: T(n,k) is the sum for all integer partitions of n of length k of the number of different multiplicities.

%H Alois P. Heinz, <a href="/A373270/b373270.txt">Rows n = 0..200, flattened</a> (first 40 rows from Olivier Gérard)

%e Array begins:

%e 1,

%e 1, 1,

%e 1, 1, 1,

%e 1, 2, 2, 1,

%e 1, 2, 4, 2, 1,

%e 1, 3, 4, 3, 2, 1,

%e 1, 3, 7, 6, 4, 2, 1,

%e 1, 4, 8, 8, 6, 4, 2, 1,

%e 1, 4, 10, 12, 10, 5, 4, 2, 1,

%e 1, 5, 12, 15, 13, 11, 6, 4, 2, 1,

%e 1, 5, 15, 21, 20, 17, 11, 6, 4, 2, 1,

%e ...

%e Example of computation:

%e T(9,3) = 10 because the partitions of 9 into 3 parts are

%e 7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,

%e the number of different multiplicities are

%e 2, 1, 1, 2, 2, 1, 1,

%e and the sum of these multiplicities is 10.

%t Flatten@Table[

%t Plus @@@

%t Table[Map[Length[Union[Length /@ Split[#]]] &,

%t IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]

%Y Cf. A373242, A373269, A373271.

%K nonn,tabl

%O 1,8

%A _Olivier Gérard_, May 29 2024