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 December 9th, 2018, 05:25 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 05:34 AM. Tags integer, nested, partititon Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Bromster Real Analysis 1 July 7th, 2014 04:04 AM Dacu Algebra 3 May 3rd, 2013 11:25 AM Dacu Algebra 31 May 2nd, 2013 08:21 AM TheLegace Applied Math 1 June 22nd, 2009 04:06 AM Dacu Calculus 0 December 31st, 1969 04:00 PM

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