A182742 Table of partitions that do not contain 1 as a part for even integers.

2, 4, 2, 3, 2, 2, 6, 3, 2, 2, 5, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 8, 4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 7, 3, 2, 2, 2, 2, 2, 2, 2, 6, 3, 3, 2, 2, 2, 2, 2, 2, 2, 5, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 10, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 3, 2, 2

Offset

1,1

Comments

This array read by antidiagonals is the main table of the shell model of partitions for even integers. Here the last sections of all even numbers are superimposed as shells of an onion. In this way many bits of information are saved.

The table is the head of the last section of partitions of an even integer when it tends to be infinite. Row n lists the parts of the n-th partition that do not contains 1 as a part.

The shell model of partitions uses this table during the filling mechanism of the head of the last section of the next even integer k. For example, in a mechanical version, the head of the last section (as a mirror) pivoting from vertical to horizontal position. Then a copy of the partitions of the integer k, listed in this table, is transmitted (or reflected) at the head (or mirror) of the last section. Finally the head (or mirror) pivots back to return to its original vertical position. And so on for all even integers.

In another version, simply a copy of the partitions of the integer k, listed in the table, are placed above the partitions of the last odd number placed in the vertical plane structure.

It appears this table is useful to know the structure of the partitions of all even integers. The same applies for odd numbers in the table of A182743. Furthermore, both tables can be unified in a three-dimensional shell model.

Links

Table of n, a(n) for n=1..95.

Example

Array begins:
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
3, 3, 2, 2, 2, 2, 2, 2, 2, 2,
6, 2, 2, 2, 2, 2, 2, 2, 2,
5, 3, 2, 2, 2, 2, 2, 2,
4, 4, 2, 2, 2, 2, 2,
8, 2, 2, 2, 2, 2,
4, 3, 3, 2, 2,
7, 3, 2, 2,
6, 4, 2,
5, 5,
10,

Crossrefs

Cf. A135010, A138121, A141285, A182730, A182736, A182740, A182743, A182746.

Column 1 give A182732. Column 2 give A182744.

Sequence in context: A279268 A279856 A054507 * A087229 A351432 A320156

Adjacent sequences: A182739 A182740 A182741 * A182743 A182744 A182745

Keyword

nonn,tabl

Author

Omar E. Pol, Nov 30 2010, Dec 01 2010, Dec 04 2010

Status

approved