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T(n,k) is the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).
4

%I #18 May 30 2024 06:58:05

%S 1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,2,2,1,1,1,1,2,1,1,2,1,1,2,1,1,1,1,2,1,

%T 1,2,2,2,2,2,2,2,2,1,1,1,1,2,1,1,2,1,1,2,2,2,2,1,2,2,2,1,2,2,2,1,1,1,

%U 1,2,1,1,2,1,1,2,2,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,1,2,2,1,1,1,1,2,1,1,2,1,1,2,2,2,1,1,1,2,2,2,2,2,1,2,1,2,2,2,2,2,2,1,2,2,2,3,2,2,1,2,2,2,2,1

%N T(n,k) is the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

%C The regular array for partitions of n of length k is A373270.

%C Row sums are A373271.

%H Olivier Gérard, <a href="/A373269/b373269.txt">Table of n, a(n) for n = 1..215307</a>

%e Array begins:

%e 1,

%e 1,1,

%e 1,1,1,

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

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

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

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

%e ...

%e T(10,34) is the first term with value 3. It corresponds to partition 3+2+2+1+1+1 of 10, which has three different multiplicities.

%t Flatten@Table[

%t Map[Length[Union[Length /@ Split[#]]] &, IntegerPartitions[n]], {n,

%t 1, 20}]

%K nonn,tabf

%O 1,10

%A _Olivier Gérard_, May 29 2024