A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

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

Offset

0,4

Comments

Row n lists the first A000041(n) terms of A141285.

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - Joerg Arndt, Sep 13 2013

Links

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

Wikiversity, Lexicographic and colexicographic order

Formula

a(n) = A061395(A334437(n)). - Gus Wiseman, May 31 2020

Example

For n = 5 the partitions of 5 in colexicographic order are:
  1+1+1+1+1
  2+1+1+1
  3+1+1
  2+2+1
  4+1
  3+2
  5
so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
Triangle begins:
  1;
  1,2;
  1,2,3;
  1,2,3,2,4;
  1,2,3,2,4,3,5;
  1,2,3,2,4,3,5,2,4,3,6;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
...

Mathematica

colex[f_, c_]:=OrderedQ[PadRight[{Reverse[f], Reverse[c]}]];
Max/@Join@@Table[Sort[IntegerPartitions[n], colex], {n, 8}] (* Gus Wiseman, May 31 2020 *)

Prog

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

Crossrefs

The sum of row n is A006128(n).

Cf. A135010, A138121, A141285, A194547, A194548, A194549.

Row lengths are A000041.

Let y be the n-th integer partition in colexicographic order (A211992):

- The maximum of y is a(n).

- The length of y is A193173(n).

- The minimum of y is A196931(n).

- The Heinz number of y is A334437(n).

Lexicographically ordered reversed partitions are A026791.

Reverse-colexicographically ordered partitions are A026792.

Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.

Reverse-lexicographically ordered partitions are A080577.

Lexicographically ordered partitions are A193073.

Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

Sequence in context: A353655 A280055 A253092 * A369320 A115452 A039676

Adjacent sequences: A194543 A194544 A194545 * A194547 A194548 A194549

Keyword

nonn,tabf

Author

Omar E. Pol, Dec 10 2011

Extensions

Definition corrected by Omar E. Pol, Sep 12 2013

a(0)=1 prepended by Andrew Howroyd, Oct 06 2025

Status

approved