A264034 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A161680(n)) is the number of integer partitions of n with weighted sum k.

1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1

Offset

0,30

Comments

Row sums give A000041.

The weighted sum is given by the sum of the rows where row i is weighted by i.

Note that the first part has weight 0. This statistic (zero-based weighted sum) is ranked by A359677, reverse A359674. Also the number of partitions of n with one-based weighted sum n + k. - Gus Wiseman, Jan 10 2023

Links

Alois P. Heinz, Rows n = 0..50, flattened

FindStat - Combinatorial Statistic Finder, Weighted size of a partition

Formula

From Alois P. Heinz, Jan 20 2023: (Start)

max_{k=0..A161680(n)} T(n,k) = A337206(n).

Sum_{k=0..A161680(n)} k * T(n,k) = A066185(n). (End)

Example

Triangle T(n,k) begins:
  1;
  1;
  1,1;
  1,1,0,1;
  1,1,1,1,0,0,1;
  1,1,1,1,1,0,1,0,0,0,1;
  1,1,1,2,1,0,2,1,0,0,1,0,0,0,0,1;
  1,1,1,2,1,1,2,1,0,1,1,1,0,0,0,1,0,0,0,0,0,1;
  1,1,1,2,2,1,2,2,1,1,1,1,1,1,0,1,1,0,0,0,0,1,0,0,0,0,0,0,1;
  ...
The a(15,31) = 5 partitions of 15 with weighted sum 31 are: (6,2,2,1,1,1,1,1), (5,4,1,1,1,1,1,1), (5,2,2,2,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1). These are also the partitions of 15 with one-based weighted sum 46. - Gus Wiseman, Jan 09 2023

Maple

b:= proc(n, i, w) option remember; expand(
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, w)+
      `if`(i>n, 0, x^(w*i)*b(n-i, i, w+1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 01 2015

Mathematica

b[n_, i_, w_] := b[n, i, w] = Expand[If[n == 0, 1, If[i < 1, 0, b[n, i - 1, w] + If[i > n, 0, x^(w*i)*b[n - i, i, w + 1]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n], Total[Accumulate[Reverse[#]]]==k&]], {n, 0, 8}, {k, n, n*(n+1)/2}] (* Gus Wiseman, Jan 09 2023 *)

Crossrefs

Row sums are A000041.

The version for compositions is A053632, ranked by A124757 (reverse A231204).

Row lengths are A152947, or A161680 plus 1.

The one-based version is also A264034, if we use k = n..n(n+1)/2.

The reverse version A358194 counts partitions by sum of partial sums.

A359677 gives zero-based weighted sum of prime indices, reverse A359674.

A359678 counts multisets by zero-based weighted sum.

Cf. A029931, A066185, A124757, A304818, A318283, A337206, A359042.

Sequence in context: A328748 A093201 A067613 * A058531 A093073 A251635

Adjacent sequences: A264031 A264032 A264033 * A264035 A264036 A264037

Keyword

nonn,tabf

Author

Christian Stump, Nov 01 2015

Status

approved