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

Author

Christian Stump, Nov 01 2015

Keywords

• nonn
• tabf

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

Examples

			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
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• FindStat - Combinatorial Statistic Finder, Weighted size of a partition

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.

Programs

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 *)

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)