This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381438 #6 Mar 02 2025 08:01:57
%S A381438 1,1,1,1,0,2,2,1,0,2,3,1,0,0,3,4,1,2,0,0,4,7,2,1,0,0,0,5,9,4,1,2,0,0,
%T A381438 0,6,13,4,4,1,0,0,0,0,8,18,6,3,2,3,0,0,0,0,10,26,9,5,2,2,0,0,0,0,0,12,
%U A381438 32,12,8,4,2,4,0,0,0,0,0,15
%N A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.
%C A381438 The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
%C A381438 Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
%C A381438 The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.
%e A381438 Triangle begins:
%e A381438 1
%e A381438 1 1
%e A381438 1 0 2
%e A381438 2 1 0 2
%e A381438 3 1 0 0 3
%e A381438 4 1 2 0 0 4
%e A381438 7 2 1 0 0 0 5
%e A381438 9 4 1 2 0 0 0 6
%e A381438 13 4 4 1 0 0 0 0 8
%e A381438 18 6 3 2 3 0 0 0 0 10
%e A381438 26 9 5 2 2 0 0 0 0 0 12
%e A381438 32 12 8 4 2 4 0 0 0 0 0 15
%e A381438 47 16 11 4 3 2 0 0 0 0 0 0 18
%e A381438 60 23 12 8 3 2 5 0 0 0 0 0 0 22
%e A381438 79 27 20 7 9 4 3 0 0 0 0 0 0 0 27
%e A381438 Row n = 9 counts the following partitions:
%e A381438 (711) (522) (333) (441) . . . . (9)
%e A381438 (6111) (4221) (3321) (81)
%e A381438 (5211) (3222) (32211) (72)
%e A381438 (51111) (22221) (222111) (63)
%e A381438 (4311) (621)
%e A381438 (42111) (54)
%e A381438 (411111) (531)
%e A381438 (33111) (432)
%e A381438 (321111)
%e A381438 (3111111)
%e A381438 (2211111)
%e A381438 (21111111)
%e A381438 (111111111)
%t A381438 egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
%t A381438 Table[Length[Select[IntegerPartitions[n],k==Last[egs[#]]&]],{n,15},{k,n}]
%Y A381438 Last column (k=n) is A000009.
%Y A381438 Row sums are A000041.
%Y A381438 Row sums without the last column (k=n) are A047967.
%Y A381438 For first instead of last part we have A116861, rank A066328.
%Y A381438 First column (k=1) is A241131 shifted right and starting with 1 instead of 0.
%Y A381438 Using Heinz numbers, this statistic is given by A381437.
%Y A381438 A122111 represents conjugation in terms of Heinz numbers.
%Y A381438 A239455 counts section-sum partitions, complement A351293.
%Y A381438 Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
%Y A381438 Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.
%Y A381438 Look-and-Say partition: A048767, A351294, A351295, A381440.
%Y A381438 Cf. A047966, A051903, A051904, A091602, A181819, A212166.
%K A381438 nonn,tabl
%O A381438 1,6
%A A381438 _Gus Wiseman_, Mar 01 2025