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).

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, 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, 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

Offset

0,6

Comments

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:

- (3,3,2,1,1) has reverse-weighted alternating sum 1*1 - 2*1 + 3*2 - 4*3 + 5*3 = 8.

- (1,2,2,3) has reverse-weighted alternating sum -1*3 + 2*2 - 3*2 + 4*1 = -1.

Links

Table of n, a(n) for n=0..93.

Example

Triangle begins:
  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  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  1  1
Row n = 6 counts the following partitions:
  k=3       k=4       k=6       k=8      k=9   k=10    k=11
--------------------------------------------------------------
  (33)      (222)  .  (6)    .  (21111)  (51)  (3111)  (411)
  (2211)              (42)
  (111111)            (321)

Mathematica

revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]], {k, 1, Length[y]}];
Table[Length[Select[IntegerPartitions[n], revaltwtsum[#]==k&]], {n, 0, 15}, {k, Floor[(n+1)/2], Ceiling[n*(n+1)/4]}]

Crossrefs

Row sums are A000041.

Column k = floor((n+1)/2) is A119620.

The unweighted version is A344612 aerated, reverse A103919.

The corresponding rank statistic is A363620, reverse A363619.

The reverse version is A363622.

A053632 counts compositions by weighted sum.

A264034 counts partitions by weighted sum, reverse A358194.

A316524 gives alternating sum of prime indices, reverse A344616.

A363624 gives weighted alternating sum of Heinz partition, reverse A363625.

Cf. A008284, A067538, A222855, A222970, A318283, A320387, A360672, A360675, A362559, A363532, A363621, A363626.

Sequence in context: A343748 A029368 A108483 * A376630 A101565 A377404

Adjacent sequences: A363620 A363621 A363622 * A363624 A363625 A363626

Keyword

nonn,tabf

Author

Gus Wiseman, Jun 15 2023

Status

approved