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A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).

%I A373241 #23 May 30 2024 06:56:55
%S A373241 0,0,0,0,1,0,0,1,0,0,0,0,1,1,0,0,0,0,0,1,1,0,0,2,0,0,1,0,0,0,1,1,0,1,
%T A373241 2,0,0,0,1,0,0,0,0,0,0,1,1,0,1,2,0,0,2,0,1,0,0,1,1,1,0,0,0,0,0,0,0,1,
%U A373241 1,0,1,2,0,1,2,0,1,0,0,2,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0,0,1,1,0,1,2,0,1,2,0,1,0,0,2,2,1,1,1,0,0,1,0,3,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0,0,0,0,0
%N A373241 T(n,k) is the difference between the number of different parts and the number of different multiplicities in the k-th partition of n in graded reverse lexicographic ordering (A080577).
%C A373241 This difference is always nonnegative.
%C A373241 The number of zero values in each row is A098859.
%C A373241 The number of ones in each row is A325244.
%C A373241 The number of positive entries in each row is A336866.
%C A373241 The corresponding regular triangle for partitions of n of length k is A373242.
%C A373241 The sum of each row is A373243.
%H A373241 Olivier Gérard, <a href="/A373241/b373241.txt">Table of n, a(n) for n = 1..215307</a>
%e A373241 The array begins
%e A373241   0
%e A373241   0,0
%e A373241   0,1,0
%e A373241   0,1,0,0,0
%e A373241   0,1,1,0,0,0,0
%e A373241   0,1,1,0,0,2,0,0,1,0,0
%e A373241   0,1,1,0,1,2,0,0,0,1,0,0,0,0,0
%e A373241   0,1,1,0,1,2,0,0,2,0,1,0,0,1,1,1,0,0,0,0,0,0
%e A373241   0,1,1,0,1,2,0,1,2,0,1,0,0,2,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0,0
%e A373241   ...
%t A373241 Flatten @ Table[
%t A373241   Map[Length[Union[#]] - Length[Union[Length /@ Split[#]]] &,
%t A373241    IntegerPartitions[n]], {n, 1, 20}]
%Y A373241 Cf. A373269 a triangle of the same shape and order for number of multiplicities.
%Y A373241 Cf. A080577, A098859, A325244, A336866, A373242, A373243.
%K A373241 nonn,tabf
%O A373241 1,24
%A A373241 _Olivier Gérard_, May 29 2024