Function Repository Resource:

ParityPairings

List all possible partitions of a list into disjoint pairs

Contributed by: Bradley Klee

ResourceFunction["ParityPairings"][items]

lists all partitionings of items into disjoint pairs matching each single item once to any other or to itself.

Details

If items has length n, each output partitioning has dimensions {k,2}, where n/2≤k≤n.

Each item from items appears exactly once as a member of a non-identical pair {l,m} or exactly twice as both members of an identical pair {m,m}.

When the items are chosen to be a range of numbers 1, 2, …, n, ResourceFunction["ParityPairings"] lists involutions of length n almost in Cycles notation, except that fixed indices are also listed as pairs {m,m}.

Examples

Basic Examples (2)

The only pairing of one item is the Identity:

In[1]:=

$ResourceFunction["ParityPairings"][{\[FormalA]}]$

Out[1]=

Two items admit two distinct pairings:

In[2]:=

$ResourceFunction["ParityPairings"][{\[FormalA], \[FormalB]}]$

Out[2]=

Alternatively, items can also be entered as integers:

In[3]:=

Out[3]=

Properties and Relations (2)

Compare with the result of the resource function Involutions:

In[4]:=

$Function[{ind}, SameQ[ResourceFunction["ParityPairings"][Range[ind]], Map[Sort, ResourceFunction["Involutions"][ind, "Cycles"] /. {x_Integer} :> {x, x}, {0, 1}] ]] /@ Range[10]$