A331581 - OEIS

A331581

Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

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

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OFFSET

1,3

COMMENTS

The first partition ranked by A080577 is (); there is no zeroth partition.

LINKS

Table of n, a(n) for n=1..97.

OEIS Wiki, Orderings of partitions

Wikiversity, Lexicographic and colexicographic order

FORMULA

a(n) = A061395(A129129(n - 1)).

EXAMPLE

The sequence of all partitions in graded reverse-lexicographic order begins as follows. The terms are the initial parts.

() (3,2) (2,1,1,1,1) (2,2,1,1,1)

(1) (3,1,1) (1,1,1,1,1,1) (2,1,1,1,1,1)

(2) (2,2,1) (7) (1,1,1,1,1,1,1)

(1,1) (2,1,1,1) (6,1) (8)

(3) (1,1,1,1,1) (5,2) (7,1)

(2,1) (6) (5,1,1) (6,2)

(1,1,1) (5,1) (4,3) (6,1,1)

(4) (4,2) (4,2,1) (5,3)

(3,1) (4,1,1) (4,1,1,1) (5,2,1)

(2,2) (3,3) (3,3,1) (5,1,1,1)

(2,1,1) (3,2,1) (3,2,2) (4,4)

(1,1,1,1) (3,1,1,1) (3,2,1,1) (4,3,1)

(5) (2,2,2) (3,1,1,1,1) (4,2,2)

(4,1) (2,2,1,1) (2,2,2,1) (4,2,1,1)

Triangle begins:

2 1

3 2 1

4 3 2 2 1

5 4 3 3 2 2 1

6 5 4 4 3 3 3 2 2 2 1

7 6 5 5 4 4 4 3 3 3 3 2 2 2 1

8 7 6 6 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 1

MATHEMATICA

revlexsort[f_, c_]:=OrderedQ[PadRight[{c, f}]];

Prepend[First/@Join@@Table[Sort[IntegerPartitions[n], revlexsort], {n, 8}], 0]

CROSSREFS

Row lengths are A000041.

Lexicographically ordered reversed partitions are A026791.

Reverse-colexicographically ordered partitions are A026792.

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

The version for compositions is A065120 or A333766.

Reverse-lexicographically ordered partitions are A080577.

Distinct parts of these partitions are counted by A115623.

Lexicographically ordered partitions are A193073.

Colexicographically ordered partitions are A211992.

Lengths of these partitions are A238966.

Cf. A036037, A048793, A063008, A066099, A129129, A185974, A228100, A228531, A334301, A334434, A334436, A334438.

Sequence in context: A211230 A049085 A193173 * A227355 A226080 A167287

Adjacent sequences: A331578 A331579 A331580 * A331582 A331583 A331584

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, May 08 2020

STATUS

approved