A182715 Triangle read by rows in which row n lists in nonincreasing order the smallest part of every partition of n.

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset

0,3

Comments

Triangle read by rows in which row n lists the smallest parts of all partitions of n in the order produced by the shell model of partitions of A138121.

Also, row n lists the "filler parts" of all partition of n. For more information see A182699.

Row n has length A000041(n). Row sums give A046746. Column 1 gives A001477. The last A000041(n-1) terms of row n are ones, n >= 1.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

Omar E. Pol, Illustration of the smallest parts of all partitions of 10 (colored blue) in the shell model of partitions

Formula

T(n,k) = A055396(A334436(n,k)). - Andrew Howroyd, Oct 06 2025

Example

For row 10, see the illustration of the link.
Triangle begins:
  0,
  1,
  2,1,
  3,1,1,
  4,2,1,1,1,
  5,2,1,1,1,1,1,
  6,3,2,2,1,1,1,1,1,1,1,
  7,3,2,2,1,1,1,1,1,1,1,1,1,1,1,
  8,4,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  9,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  ...

Prog

(PARI) Row(n)=[if(!#p, 0, p[1]) | p<-vecsort(partitions(n), , 4)]
{ for(n=0, 9, print(Row(n))) } \\ Andrew Howroyd, Oct 06 2025

Crossrefs

Cf. A026807, A055396, A135010, A138121, A141285, A182699, A182709, A334436.

Mirror of triangle A196931.

Sequence in context: A062534 A143349 A335123 * A088425 A141294 A174557

Adjacent sequences: A182712 A182713 A182714 * A182716 A182717 A182718

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 01 2010

Extensions

Name simplified and more terms from Omar E. Pol, Oct 21 2011

Status

approved