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December 9th, 2018, 05:25 AM   #1
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Nested integer partititon

Here is some problem that is related to Combinatorics and Number Theory.

Please observe the following diagram of the natural numbers 1 to 4:




This diagram represents the transition from multiplicity to addition under a given natural number > 0, such that multiplicity is done among 1's that do not have unique identities (therefore they can be summed by a single operation) and addition is done among 1's that have unique identities (therefore they can't be summed by a single operation (unless there is only a single 1)).


Here are the transitions from multiplicity to addition under the given natural numbers 1 to 4:

1: (+1) (its own uniqueness (therefore no multiplication))


2: (1*2), ((+1)+1)


3: (1*3), ((1*2)+1), (((+1)+1)+1)


4: (1*4), ((1*2)+1*2), (((+1)+1)+1*2), ((1*2)+(1*2)), (((+1)+1)+(1*2)), (((+1)+1)+((+1)+1)), ((1*3)+1), (((1*2)+1)+1), ((((+1)+1)+1)+1)


My question is:

How can we define an equation that returns the number of these nested forms under any given natural number > 0?


My question is about nested integer partition, which is an extension of integer partition ( https://en.wikipedia.org/wiki/Partition_(number_theory) ).

My nested integer partition is defined by the transition from symmetry (no 1's under a given n>1 have a unique "name" (order is impossible)) to asymmetry (all 1's under a given n>1 have a unique "name" (order is fully possible)).

For example: in case of n=4, the most symmetrical state is defined as (1*4) and the most asymmetrical state is defines as ((((+1)+1)+1)+1).

Diagrams of natural numbers 1 to 6 are seen here:



Last edited by doronshadmi; December 9th, 2018 at 05:34 AM.
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