

A194546


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


8



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


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 colexicographic, see example.  Joerg Arndt, Sep 13 2013


LINKS

Table of n, a(n) for n=1..96.
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 *)


CROSSREFS

The sum of row n is A006128(n).
Cf. A135010, A138121, A141285, A194547, A194548, A194549.
Row lengths are A000041.
Let y be the nth 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.
Reversecolexicographically ordered partitions are A026792.
Reversed partitions in AbramowitzStegun order (sum/length/lex) are A036036.
Reverselexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.
Sequence in context: A285730 A280055 A253092 * A115452 A039676 A242359
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


STATUS

approved



