Permutations and partitions in 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.

Partitions of a finite set and partitions of an integer are also often ordered lexicographically, but the sequences and  define an infinite order.

These infinite orderings make it possible to define some mappings as sequences:

Each permutation corresponds to a set partition, and more importantly to an integer partition 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 - 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, for set partitions the non-singleton blocks and for integer partitions the non-one addends.

And of course they can be assigned the total number of elements in these "blocks":, and a staircase like sequence that is too boring to be in the OEIS.

Equalities
For simplicity's sake it is assumed that and  have index 0 like all the other sequences, although at the moment they have index 1 in the OEIS.