A386832 Triangle read by rows: T(n,k) is the number of the sets of partitions of n with length k grouped by largest part.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 3, 3, 2, 1, 1, 1, 4, 4, 4, 3, 2, 1, 1, 1, 4, 5, 4, 4, 3, 2, 1, 1, 1, 5, 5, 5, 5, 4, 3, 2, 1, 1, 1, 5, 6, 6, 5, 5, 4, 3, 2, 1, 1, 1, 6, 7, 7, 6, 6, 5, 4, 3, 2, 1, 1, 1, 6, 7, 7, 7, 6, 6, 5, 4, 3, 2, 1, 1, 1, 7, 8, 8, 8, 7, 7, 6, 5, 4, 3, 2, 1, 1, 1, 7, 9, 9, 9, 8, 7, 7, 6, 5, 4, 3, 2, 1, 1, 1, 8, 9, 10, 9, 9, 8, 8, 7, 6, 5, 4, 3, 2, 1, 1

Offset

1,8

Comments

Row sums are equal to (n^2 + 3*n)/2 - Sum_{k=1..n} ceiling(n/k) which is conjectured to be equal to A309097: "Number of partitions of n avoiding the partition (4,2,1)." (cf. Geoffrey Caveney link).

Conjecture is proven true by Kevin Ryde (see A309097).

Links

Table of n, a(n) for n=1..136.

Wouter Meeussen, proofs by Geoffrey Caveney.

Formula

T(n,k) = T(n-1,k-1) + 1-[n-k+1|n] with T(n,0)=1.

Example

For n=7,
  1 : (7)
  3 : (6,1)             (5,2)         (4,3)
  3 : (5,1,1)           (4,2,1)       {(3,3,1),(3,2,2)}
  3 : (4,1,1,1)         (3,2,1,1)     (2,2,2,1)
  2 : (3,1,1,1,1)       (2,2,1,1,1)
  1 : (2,1,1,1,1,1)
  1 : (1,1,1,1,1,1,1)
partitions (3,3,1) and (3,2,2) both belong to the set of partitions of n=7 with k=length=3 and largest part 3; so T(7,3)=3, counting (5,1,1), (4,2,1) and ((3,3,1),(3,2,2)).
Table starts as:
  1
  1, 1
  1, 1, 1
  1, 2, 1, 1
  1, 2, 2, 1, 1
  1, 3, 3, 2, 1, 1
  1, 3, 3, 3, 2, 1, 1
  1, 4, 4, 4, 3, 2, 1, 1
  1, 4, 5, 4, 4, 3, 2, 1, 1
  1, 5, 5, 5, 5, 4, 3, 2, 1, 1

Mathematica

Table[Length/@Table[Map[Length, GatherBy[IntegerPartitions[n, {k}], First], {1}], {k, n}], {n, 10}];
(* or *) Clear[T]; T[_, 0]:=1; T[n_, k_]:=T[n, k]=T[n-1, k-1]+1-Boole[Divisible[n, n-k+1]]; Table[T[n, n-k], {n, 0, 9}, {k, 0, n}]

Crossrefs

Cf. A309097.

Sequence in context: A166240 A219347 A114087 * A215521 A008284 A114088

Adjacent sequences: A386829 A386830 A386831 * A386833 A386834 A386835

Keyword

nonn,tabl

Author

Wouter Meeussen, Sep 03 2025

Status

approved