File:Set partitions with no singletons until place n.svg

This table shows the 52 partitions of a 5-element set
and the 41 non-singleton partitions of a 6-element set
(represented by logical matrices of equivalence relations)
ordered by the position of singletons.

Singletons are 1-element sets, and in the matrices they are represented by light gray squares on the main diagonal.

The rows have values n from 1 to 6.

In the cells of the left column are the non-singleton partitions of an n-element set -
or in other terms: The partitions with no singleton until place n

In the cells of the right column are the partitions of an n-element set with singletons
that can't be described as partitions of a smaller set, and are thus not shown before.

The number of partitions in the cells of the left column is 0, 1, 1, 4, 11, 41 ...
This is sequence A =  A000296 from entry A(1).
It has the feature that A(n) + A(n+1) = B(n), where B is the sequence of Bell numbers ( A000110).

The highlighted partitions in the small tables below the main one show that
the number of singleton containing partitions of a set with n elements
equals