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December 9th, 2018, 06:25 AM  #1 
Newbie Joined: Apr 2011 Posts: 19 Thanks: 0  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 06:34 AM. 

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