# Partition Diagrams

The similarity between adding partitions to set partitions and adding
items to partitions of set partitions leads to the idea of recursively
or hierarchically partitioning a set. Schroeder diagrams will be used
to represent the process of recursive partitioning. The combinatoric structure associated with the problem of
recursive partitioning is the combinatoric structure known as
Schroeder's Fourth Problem or hierarchies.

A new rule needs to be added for recursive partitioning, stating that
internal nodes must have two or more child nodes. The absence of this
rule would allow an internal node with one child node to be recursively
partitioned indefinitely, creating an infinite number of trees. A new
type of diagram is introduced, the Schroeder diagram. The hexagon Bell
number diagram in Figure 2 has two partitions of three items each, but
a group of three items may be further partitioned in four ways. A tree
structure of Bell number diagrams allows recursive partitioning to be
displayed in a less visually confusing manner that displaying it with a
standard tree graph.

A Schroeder diagram represents the process of recursive partitioning in
the form of a tree with Bell number diagrams for nodes. Bell number
diagrams containing only one partition are forbidden, as in Figure 1.
The absence of this rule would allow an internal node with one child
node to be recursively partitioned indefinitely, creating an infinite
number of trees. Later it will be shown that these diagrams are
forbidden because they are left hand values. Figure 1 represent the
third, fourth, and fifth derivatives of a recursive function.

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Figure 1 - Examples of Bell Number Diagrams Forbidden in Schroeder Diagrams

Partitions with 1 or 2 items are not displayed as their own separate
Bell number diagram since no additional information would be
forthcoming; a single item can't be sub-partitioned, and two items can
only be recursively partitioned in one way due to the prohibition
against placing all items in a single partition. All Bell number diagram partitions of 3 or more items have a second
representation as child node Bell number diagram containing the same
number of items as the original partition. Child nodes are sorted from left to right in chronological order; a
child node containing the first label would be to the left of the other
child nodes. Items and partitions are commingled in Schroeder diagrams rather that
have partitions with only one item. A "naked" single item replaces
partitions containing a single item.

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Figure 2 - The Schroeder diagram and tree graph of {{{1, 5}, 3}, {{2, 4}, 6}}

Diagrams of unlabeled combinatoric structures are placed in a circle to
indicate that they serve as a grouping of their labeled versions. The
unlabeled structures then can be used to index the labeled structures.
Fa`a di Bruno's Formula uses the enumeration of the integer partitions
to form the index for the summation that gives the different
derivatives of composite functions. The Ferrers diagrams below visually
enumerate the integer partitions of the number four.

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Figure 3 - The Ferrers diagrams of the number four

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Figure 4 -
Unlabeled Bell Number Diagrams

Enumerates integer partitions; a radial Ferrers diagram. This serves as a building block for the next type of diagram.

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Figure 5 -
Unlabeled Schroeder Diagrams

Enumerates unlabeled hierarchies and serves as an index for labeled hierarchies.

## Bell Number Diagrams

## Unlabeled Bell Number Diagrams

Enumerates integer partitions, a
radial Ferrers diagram.

## Unlabeled Schroeder Diagrams

Enumerates unlabeled hierarchies

## Schroeder Diagrams

The entries after hier(2) are grouped together with other structures
identical except for their labeling and preceded by the unlabeled
diagram serving to index the entire group.

### hier(1)

### hier(2)

### hier(3)

### hier(4)

### hier(5)