User:Tilman Piesk/Lists with offset 0

Lists in the OEIS usually have offset 1.
An example is A018900, which is the list of 2-element sets, interpreted as binary numbers.
All entries of this sequence can be ordered in an array (tabl)
where entries in the same row belong to the same equivalence class.
(In this case also the columns can be seen as equivalence classes.)

However, there are lists that are different from this one in an essential point:
Their first element belongs to an equivalence class with only this element,
while all other elements can be ordered in equivalence classes with an infinite number of elements.
So all entries but the first one can be ordered in an array
where entries in the same row belong to the same equivalence class.

Such sequences could also be defined without the first element
by adding something like "non-identity" or "nontrivial" in the definition.

For some reasons it is practical to enumerate this sui generis element with index number 0 rather than 1.

Permutations and partitions

The probably best known list with an offset 0 is that of finite permutations (compare A195663 or A055089),
where the index number is given by the factorial number (the reflected inversion vector).

The permutations may be ordered in all kinds of equivalence classes - by number of cycles, by corresponding integer partition, by inversion number or whatever. But the identity permutation will always be in an equivalence class with no other elements.

Partitions of sets and integers belong to the same branch of mathematics:
Each permutation can be assigned a set partition, each set partition can be assigned an integer partition.

The identity permutation corresponds to the only-singletons set partition, which corresponds to the only-ones integer partition,
so the lists of partitions have the same kind of sui generis element.

Such elements do not appear in the following infinite arrays:

Permutations corresponding to the same integer partition ordered in the same row of an array	Set partitions corresponding to the same integer partition ordered in the same row of an array
(compare Set partitions 5)

If the integer partitions were enumerated from 1 instead of 0,
the rows had to be enumerated from 2 instead 1, which would neither be usual nor practical.

If the permutations or set partitions were enumerated from 1 instead of 0,
the arrays wouldn't contain all positive integers, but the integers from 2.
The array flattened to a sequence would than be a permutation of the integers from 2 instead of the positive integers.
As no one permutes integers from 2 this would also be not useful at all.