A196931 - OEIS

A196931

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

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

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OFFSET

0,4

COMMENTS

If n >= 1, row n lists the smallest parts of every partition of n in the order produced by the shell model of partitions of A135010, hence row n lists the parts of the last section of the set of partitions of n, except the emergent parts (See A182699).

Row n has length A000041(n). Row sums give A046746. Right border of triangle gives A001477. Row n starts with A000041(n-1) ones, n >= 1.

LINKS

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

FORMULA

T(n,k) = A055396(A334437(n,k)). - Andrew Howroyd, Oct 06 2025

EXAMPLE

Written as a triangle:

1,2,

1,1,3,

1,1,1,2,4,

1,1,1,1,1,2,5,

1,1,1,1,1,1,1,2,2,3,6

1,1,1,1,1,1,1,1,1,1,1,2,2,3,7,

1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,8,

...

PROG

(PARI) Row(n)=[if(!#p, 0, p[1]) | p<-vecsort(partitions(n))]

{ for(n=0, 9, print(Row(n))) } \\ Andrew Howroyd, Oct 06 2025

CROSSREFS

Mirror of triangle A182715.

Cf. A000041, A046746, A055396, A135010, A138121, A141285, A182699, A183152, A193827, A196930, A334437.

Sequence in context: A138138 A230440 A283495 * A175465 A080209 A347171

Adjacent sequences: A196928 A196929 A196930 * A196932 A196933 A196934

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Oct 21 2011

STATUS

approved