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A373242 T(n,k) is the sum for all integer partitions of n of length k of the difference between the number of different parts and the number of different multiplicities.

%I A373242 #18 May 30 2024 06:59:11
%S A373242 0,0,0,0,1,0,0,1,0,0,0,2,0,0,0,0,2,2,1,0,0,0,3,2,1,0,0,0,0,3,4,3,1,0,
%T A373242 0,0,0,4,6,4,2,2,0,0,0,0,4,8,8,5,1,1,0,0,0,0,5,10,10,7,2,1,1,0,0,0,0,
%U A373242 5,14,16,12,8,3,2,1,0,0,0,0,6,16,20,17,8,6,2,1,1,0,0,0,0,6,20,29,25,16,10,5,2,1,1,0,0,0,0,7,24,35,36,27,14,7,6,3,1,1,0,0,0
%N A373242 T(n,k) is the sum for all integer partitions of n of length k of the difference between the number of different parts and the number of different multiplicities.
%C A373242 The corresponding irregular triangle (one entry for each partition of n) is A373241.
%C A373242 The sum of each row is A373243.
%C A373242 The corresponding triangle for sum of number of different parts is A092905.
%C A373242 The corresponding triangle for sum of number of different multiplicities is A373271.
%H A373242 Olivier Gérard, <a href="/A373242/b373242.txt">Table of n, a(n) for n = 1..820</a>
%e A373242 Array begins:
%e A373242   0
%e A373242   0,0
%e A373242   0,1,0
%e A373242   0,1,0,0
%e A373242   0,2,0,0,0
%e A373242   0,2,2,1,0,0
%e A373242   0,3,2,1,0,0,0
%e A373242   0,3,4,3,1,0,0,0
%e A373242   0,4,6,4,2,2,0,0,0
%e A373242   0,4,8,8,5,1,1,0,0,0
%e A373242   ...
%e A373242 Example of computation:
%e A373242 T(9,3) = 6 because the partitions of 9 into 3 parts are
%e A373242   7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3,
%e A373242 the numbers of different parts are
%e A373242   2, 3, 3, 2, 2, 3, 1,
%e A373242 the numbers of different multiplicities are
%e A373242   2, 1, 1, 2, 2, 1, 1,
%e A373242 the differences between them are
%e A373242   0, 2, 2, 0, 0, 2, 0,
%e A373242 and the sum of these differences is 6.
%t A373242 Flatten[Table[
%t A373242   Plus @@@
%t A373242    Table[Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
%t A373242      IntegerPartitions[n, {k}]], {k, 1, n}], {n, 1, 20}]]
%K A373242 nonn,tabl
%O A373242 1,12
%A A373242 _Olivier Gérard_, May 29 2024