A338411 Irregular triangle read by rows enumerating nonempty finite ballot sequences by increasing length and then by lexicographical order.

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

Offset

1,5

Comments

Ballot sequences B have positive terms, and for any finite prefix P of B and any k > 0, the number of occurrences of k in P is greater than or equal to the number of occurrences of k+1 in P.

Shuffling two ballot sequences gives another ballot sequence.

The ordinal transform of a ballot sequence is also a ballot sequence.

The ordinal transform restricted to the set of ballot sequences is a self-inverse permutation.

For any n > 0:

- there are A000085(n) rows with n terms,

- the first row with n terms has only 1's,

- the last row with n terms equals [1, 2, ..., n].

Links

Rémy Sigrist, Table of n, a(n) for n = 1..8379

OEIS Wiki, Ordinal transform

Rémy Sigrist, PARI program for A338411

Example

Table begins:
     1:   [1]
     2:   [1, 1]
     3:   [1, 2]
     4:   [1, 1, 1]
     5:   [1, 1, 2]
     6:   [1, 2, 1]
     7:   [1, 2, 3]
     8:   [1, 1, 1, 1]
     9:   [1, 1, 1, 2]
    10:   [1, 1, 2, 1]
    11:   [1, 1, 2, 2]
    12:   [1, 1, 2, 3]
    13:   [1, 2, 1, 1]
    14:   [1, 2, 1, 2]
    15:   [1, 2, 1, 3]
    16:   [1, 2, 3, 1]
    17:   [1, 2, 3, 4]

Prog

(PARI) See Links section.

Crossrefs

Cf. A000085.

Sequence in context: A286520 A320105 A120698 * A366779 A326775 A349410

Adjacent sequences: A338408 A338409 A338410 * A338412 A338413 A338414

Keyword

nonn,tabf

Author

Rémy Sigrist, Oct 25 2020

Status

approved