cp's OEIS Frontend

A363623 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-weighted alternating sum k (leading and trailing 0's omitted).

%I A363623 #4 Jun 15 2023 20:06:48
%S A363623 1,1,1,1,1,2,2,0,1,2,2,1,1,1,1,1,3,1,0,3,0,1,1,1,1,3,2,0,3,1,2,0,1,0,
%T A363623 1,2,5,1,0,3,1,2,2,2,1,1,0,1,0,1,2,5,3,0,4,2,2,0,3,2,1,3,0,0,1,0,1,1,
%U A363623 1,1,7,2,0,4,1,5,2,3,1,3,0,2,3,1,2,1,0,0,1,0,1,1,1,1
%N A363623 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-weighted alternating sum k (leading and trailing 0's omitted).
%C A363623 We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) i * y_{k-i+1}. For example:
%C A363623 - (3,3,2,1,1) has reverse-weighted alternating sum 1*1 - 2*1 + 3*2 - 4*3 + 5*3 = 8.
%C A363623 - (1,2,2,3) has reverse-weighted alternating sum -1*3 + 2*2 - 3*2 + 4*1 = -1.
%e A363623 Triangle begins:
%e A363623   1
%e A363623   1
%e A363623   1  1
%e A363623   1  2
%e A363623   2  0  1  2
%e A363623   2  1  1  1  1  1
%e A363623   3  1  0  3  0  1  1  1  1
%e A363623   3  2  0  3  1  2  0  1  0  1  2
%e A363623   5  1  0  3  1  2  2  2  1  1  0  1  0  1  2
%e A363623   5  3  0  4  2  2  0  3  2  1  3  0  0  1  0  1  1  1  1
%e A363623 Row n = 6 counts the following partitions:
%e A363623   k=3       k=4       k=6       k=8      k=9   k=10    k=11
%e A363623 --------------------------------------------------------------
%e A363623   (33)      (222)  .  (6)    .  (21111)  (51)  (3111)  (411)
%e A363623   (2211)              (42)
%e A363623   (111111)            (321)
%t A363623 revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
%t A363623 Table[Length[Select[IntegerPartitions[n],revaltwtsum[#]==k&]],{n,0,15},{k,Floor[(n+1)/2],Ceiling[n*(n+1)/4]}]
%Y A363623 Row sums are A000041.
%Y A363623 Column k = floor((n+1)/2) is A119620.
%Y A363623 The unweighted version is A344612 aerated, reverse A103919.
%Y A363623 The corresponding rank statistic is A363620, reverse A363619.
%Y A363623 The reverse version is A363622.
%Y A363623 A053632 counts compositions by weighted sum.
%Y A363623 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363623 A316524 gives alternating sum of prime indices, reverse A344616.
%Y A363623 A363624 gives weighted alternating sum of Heinz partition, reverse A363625.
%Y A363623 Cf. A008284, A067538, A222855, A222970, A318283, A320387, A360672, A360675, A362559, A363532, A363621, A363626.
%K A363623 nonn,tabf
%O A363623 0,6
%A A363623 _Gus Wiseman_, Jun 15 2023