Permutations and partitions in the OEIS
|This article references|
sequences from the OEIS.
Permutations of a finite set are often sorted lexicographically, but reverse colexicographical order gives an infinite order of finite permutations.
(In this order the inversion vector of the n-th permutation interpreted as a factorial number is n.)
The finite permutations are stored as rows of a triangle in A055089.
These infinite orderings make it possible to define some mappings as sequences:
Each permutation corresponds to a set partition (A249618), and more importantly to an integer partition (A198380) called its cycle type.
(A set partition corresponds only to a finite number of permutations, but each cycle type > 0 corresponds to an infinite number of permutations.)
Each set partition corresponds to an integer partition (A249617) - and each integer partition > 0 to an infinite number of set partitions.
Furthermore each permutation and partition can be assigned a number that for simplicity's sake is called "blocks" in the diagram:
For permutations these are the cycles except fixed points (A055090), for set partitions the non-singleton blocks (A249615) and for integer partitions the non-one addends (A194548).
|pe2ip = sp2ip( pe2sp )||A198380 = A249617( A249618 )|
|pe2bl = sp2bl( pe2sp ) = ip2bl( sp2ip( pe2sp ) )||A055090 = A249615( A249618 ) = A194548( A249617( A249618 ) )|
|sp2bl = ip2bl( sp2ip )||A249615 = A194548( A249617 )|
|pe2el = sp2el( pe2sp ) = ip2el( sp2ip( pe2sp ) )||A055093 = A249616( A249618 ) = ip2el( A249617( A249618 ) )|
|sp2el = ip2el( sp2ip )||A249616 = ip2el( A249617 )|