A241596 Partitions listed by alternately incrementing each part and appending a 1.

1, 2, 11, 3, 22, 21, 111, 4, 33, 32, 222, 31, 221, 211, 1111, 5, 44, 43, 333, 42, 332, 322, 2222, 41, 331, 321, 2221, 311, 2211, 2111, 11111, 6, 55, 54, 444, 53, 443, 433, 3333, 52, 442, 432, 3332, 422, 3322, 3222, 22222, 51, 441, 431, 3331, 421, 3321, 3221, 22221, 411, 3311, 3211, 22211, 3111, 22111, 21111, 111111

Offset

1,2

Comments

Start with S_0 = {1}.

Thereafter, S_{n+1} consists of the partitions in S_n with all parts incremented by 1, together with all partitions in S_n with an additional part of 1.

From Franklin T. Adams-Watters, May 19 2014:

a(n) can be defined in terms of the binary expansion of n. Start with the partition [1]. Now process the bits of n from right to left, excluding the leading 1. For a zero bit, increase each number in the partition by 1; for a one bit, add a part of size 1. For example, for n=11, binary 1011, we get 1 -> 11 -> 111 -> 222 = a(11).

Row n consists of all partitions with hook size (maximum part + number of parts - 1) equal to n.

This sequence will eventually fail because digits greater than 9 are needed.

References

Arie Groeneveld, Posting to Sequence Fans List, May 19 2014

Links

Alois P. Heinz, Rows n = 1..9, flattened

Example

The partitions appear in the following order:
S_0 = 1,
S_1 = 2, 11,
S_2 = 3, 22, 21, 111,
S_3 = 4, 33, 32, 222, 31, 221, 211, 1111,
S_4 = 5, 44, 43, 333, 42, 332, 322, 2222, 41, 331, 321, 2221, 311, 2211, 2111, 11111,
...

Maple

b:= proc(n) option remember; `if`(n=1, [[1]],
      [map(x-> map(y-> y+1, x), b(n-1))[],
       map(x-> [x[], 1], b(n-1))[]])
    end:
T:= n-> map(x-> parse(cat(x[])), b(n))[]:
seq(T(n), n=1..6);

Crossrefs

See A242628 for another version of this list of partitions.

Cf. A125106, A240837, A112531, A241597 (compositions).

Sequence in context: A068743 A322761 A113736 * A365769 A339616 A073331

Adjacent sequences: A241593 A241594 A241595 * A241597 A241598 A241599

Keyword

nonn,tabf,base

Author

N. J. A. Sloane, May 19 2014

Extensions

Typos corrected by Alois P. Heinz, Sep 25 2015

Status

approved