login
A338411
Irregular triangle read by rows enumerating nonempty finite ballot sequences by increasing length and then by lexicographical order.
1
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
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
KEYWORD
nonn,tabf
AUTHOR
Rémy Sigrist, Oct 25 2020
STATUS
approved