Preferential arrangements of set partitions

This article is foremost a supplement to Formulas in predicate logic, because a formula with an n-place predicate is essentially a PA of an n-set together with a Boolean value.

Intuitively a preferential arrangement (PA) of a partition of a set is a ranking of the partition's blocks in a hierarchy (where several blocks may be in the same level of the hierarchy).
(As there is no risk of confusion, a PA of a partition of a set is simply called a PA of a set here.)

Example {1,2}{3}{4}

The blocks of the partition {1,2}{3}{4} can be assigned to levels in the following ways - with the colon ":" as a separator between two levels:
{1,2}{3}{4}
{1,2}{3}:{4}     {4}:{1,2}{3}
{1,2}{4}:{3}     {3}:{1,2}{4}
{3}{4}:{1,2}     {1,2}:{3}{4}
{1,2}:{3}:{4}     {1,2}:{4}:{3}       {3}:{1,2}:{4}     {3}:{4}:{1,2}       {4}:{1,2}:{3}     {4}:{3}:{1,2}

The above syntax is used on the OEIS pages linked below. Here this will be written in the following way, to harmonize with abbreviations in Formulas in predicate logic:
(12)34
(12)3|4     4|(12)3
(12)4|3     3|(12)4
34|(12)     (12)|34
(12)|3|4     (12)|4|3       3|(12)|4     3|4|(12)       4|(12)|3     4|3|(12)

Sometimes (especially in variables - like here) parentheses are not allowed, so something like this may be the solution:
12_3_4
12_3|4     4|12_3
12_4|3     3|12_4
3_4|12     12|3_4
12|3|4     12|4|3       3|12|4     3|4|12       4|12|3     4|3|12

More abstractly: a PA of the set {1...n} is a partition of {1...n} with k parts, a partition of {1...k} with i parts and a permutation of (1...i)

- or in other terms: a partition of {1...n} with k parts and an ordered partition of {1...k}.
(So the PA correspnding to the finest partition 1∣...∣n are simply the ordered partitions of {1...n}.)

In the following illustrations for n=3 and n=4 every combination of a red partition, a green partition over it and a blue permutation left of it is a PA:

n=3

The 2*23 = 46 3-place formulas in a Hasse diagram

The 23 PA of the set {1,2,3}

Examples

( 1∣2∣3, 13∣2, 21 ) = 2∣13

The 1*1 + 2*3 + 6*1 = 13 PA that correspond to the finest partition 1∣2∣3

(The ordered partitions of {1,2,3} in the permutohedron-like Cayley graph of S₃)

( 13∣2, 1∣2, 21 ) = 2∣(13)

Predicate logic
Formulas corresponding to 2∣13: a2 e13 e2 a13	The 2*13 = 26 formulas that correspond to the finest partition 1∣2∣3 (Hasse diagram of the lattice of formulas without coinciding variables)
Formulas corresponding to 2∣(13): a2 e(13) e2 a(13)

n=4

The 175 PA of the set {1,2,3,4}

Examples

( 1∣2∣3∣4, 13∣2∣4, 312 ) = 4∣13∣2

The 1*1 + 2*7 + 6*6 + 24*1 = 75 PA that correspond to the finest partition 1∣2∣3∣4

(The ordered partitions of {1,2,3,4} in the permutohedron-like Cayley graph of S₄)

( 14∣2∣3, 12∣3, 21 ) = 3∣(14)2

Predicate logic
Formulas corresponding to 4∣13∣2: a4 e13 a2 e4 a13 e2	The 2*75 = 150 formulas that correspond to the finest partition 1∣2∣3∣4 (Hasse diagram of the lattice of formulas without coinciding variables)
Formulas corresponding to 3∣(14)2: a3 e(14)2 e3 a(14)2

Number of PA

There are A083355(n) PA of an n-set.
The illustrations above indicate two ways to order them by a second property:
A232598(n,k) = S2(n,k) * oB(k) is the number of n-set PA with k blocks.
A233357(n,i) = S2²(n,i) * i! with S2²= A039810 is the number of n-set PA with i levels.

          A232598                                   A233357                                     A083355
 
          k = 1    2     3      4      5            i = 1     2      3      4       5              sums
n
1             1                                         1                                             1
2             1    3                                    2     2                                       4
3             1    9    13                              5    12      6                               23
4             1   21    78     75                      15    64     72     24                       175
5             1   45   325    750    541               52   350    660    480     120              1662