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.

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, 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, 32, 12, 8, 4, 2, 4, 0, 0, 0, 0, 0, 15

Offset

1,6

Comments

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).

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.

The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

Links

Table of n, a(n) for n=1..78.

Example

Triangle begins:
   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  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
  32 12  8  4  2  4  0  0  0  0  0 15
  47 16 11  4  3  2  0  0  0  0  0  0 18
  60 23 12  8  3  2  5  0  0  0  0  0  0 22
  79 27 20  7  9  4  3  0  0  0  0  0  0  0 27
 Row n = 9 counts the following partitions:
  (711)        (522)    (333)     (441)  .  .  .  .  (9)
  (6111)       (4221)   (3321)                       (81)
  (5211)       (3222)   (32211)                      (72)
  (51111)      (22221)  (222111)                     (63)
  (4311)                                             (621)
  (42111)                                            (54)
  (411111)                                           (531)
  (33111)                                            (432)
  (321111)
  (3111111)
  (2211111)
  (21111111)
  (111111111)

Mathematica

egs[y_]:=If[y=={}, {}, Table[Total[Select[Union[y], Count[y, #]>=i&]], {i, Max@@Length/@Split[y]}]];
Table[Length[Select[IntegerPartitions[n], k==Last[egs[#]]&]], {n, 15}, {k, n}]

Crossrefs

Last column (k=n) is A000009.

Row sums are A000041.

Row sums without the last column (k=n) are A047967.

For first instead of last part we have A116861, rank A066328.

First column (k=1) is A241131 shifted right and starting with 1 instead of 0.

Using Heinz numbers, this statistic is given by A381437.

A122111 represents conjugation in terms of Heinz numbers.

A239455 counts section-sum partitions, complement A351293.

Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.

Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.

Look-and-Say partition: A048767, A351294, A351295, A381440.

Cf. A047966, A051903, A051904, A091602, A181819, A212166.

Sequence in context: A246785 A084934 A125927 * A285864 A092869 A029337

Adjacent sequences: A381435 A381436 A381437 * A381439 A381440 A381441

Keyword

nonn,tabl

Author

Gus Wiseman, Mar 01 2025

Status

approved