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Pyth, 10 9 bytes

{SMlMM./U

Not too sure if this isn't cheating, but the rules only said one cannot use integer partition (it's not stated clearly in the question itself, but a comment by OP in the question says integer partition). I'm using string list partition, which makes slices of the list that concatenate up to the "mother" list. I believe I have to thank @Maltysen for the idea of using lists rather than strings.

n=15 takes less than one second on my machine.

In dataflow pseudocode:

              input       // initial data
        U     range       // makes a list of length equal to input
      ./      partition   // partitions string
   lMM        length      // length of each substring in each way to partition
 SM           sort        // sort each way to partition
{             deduplicate // removes all duplicate after sorting
              print       // implicit, output final result

Try it online here.

Pyth, 10 9 bytes

{SMlMM./U

Not too sure if this isn't cheating, but the rules only said one cannot use integer partition (it's not stated clearly in the question itself, but a comment by OP in the question says integer partition). I'm using string list partition, which makes slices of the list that concatenate up to the "mother" list. I believe I have to thank @Maltysen for the idea of using lists rather than strings.

n=15 takes less than one second on my machine.

In dataflow pseudocode:

              input       // initial data
        U     range       // makes a list of length equal to input
      ./      partition   // partitions string
   lMM        length      // length of each substring in each way to partition
 SM           sort        // sort each way to partition
{             deduplicate // removes all duplicate after sorting
              print       // implicit, output final result

Try it online here.

Pyth, 10 bytes

{SMlMM./*N

Not too sure if this isn't cheating, but the rules only said one cannot use integer partition (it's not stated clearly in the question itself, but a comment by OP in the question says integer partition). I'm using string partition, which makes substrings that concatenate up to the "mother" string.

n=15 takes less than one second on my machine.

In dataflow pseudocode:

              input       // initial data
        *N    times '"'   // makes a string of length equal to input
      ./      partition   // partitions string
   lMM        length      // length of each substring in each way to partition
 SM           sort        // sort each way to partition
{             deduplicate // removes all duplicate after sorting
              print       // implicit, output final result

Try it online here.

Pyth, 10 9 bytes

{SMlMM./U

Not too sure if this isn't cheating, but the rules only said one cannot use integer partition (it's not stated clearly in the question itself, but a comment by OP in the question says integer partition). I'm using string list partition, which makes slices of the list that concatenate up to the "mother" list. I believe I have to thank @Maltysen for the idea of using lists rather than strings.

n=15 takes less than one second on my machine.

In dataflow pseudocode:

              input       // initial data
        U     range       // makes a list of length equal to input
      ./      partition   // partitions string
   lMM        length      // length of each substring in each way to partition
 SM           sort        // sort each way to partition
{             deduplicate // removes all duplicate after sorting
              print       // implicit, output final result

Try it online here.

Pyth, 11 bytes

{SMlMM./*N

Not too sure if this isn't cheating, but the rules only said one cannot use integer partition (it's not stated clearly in the question itself, but a comment by OP in the question says integer partition). I'm using string partition, which makes substrings.

In dataflow pseudocode:

              input       // initial data
        *N    times '"'   // makes a string of length equal to input
      ./      partition   // partitions string
   lMM        length      // length of each substring in each way to partition
 SM           sort        // sort each way to partition
{             deduplicate // removes all duplicate after sorting
              print       // implicit, output final result

Try it online here.

Pyth, 10 bytes

{SMlMM./*N

Not too sure if this isn't cheating, but the rules only said one cannot use integer partition (it's not stated clearly in the question itself, but a comment by OP in the question says integer partition). I'm using string partition, which makes substrings that concatenate up to the "mother" string.

n=15 takes less than one second on my machine.

In dataflow pseudocode:

              input       // initial data
        *N    times '"'   // makes a string of length equal to input
      ./      partition   // partitions string
   lMM        length      // length of each substring in each way to partition
 SM           sort        // sort each way to partition
{             deduplicate // removes all duplicate after sorting
              print       // implicit, output final result

Try it online here.