Question

Prove that the number of partitions of the positive integer $n$ into parts each of which is at most 2 equals $\lfloor n / 2\rfloor+1$. (Remark: There is a formula, namely the nearest integer to $\frac{(n+3)^{2}}{12}$, for the number of partitions of $n$ into parts each of which is at most 3 but it is much more difficult to prove. There is also one for partitions with no part more than 4 , but it is even more complicated and difficult to prove.)

Name: Prove that the number of partitions of the positive integer n into parts each of which is at most 2 equals ⌊ n / 2⌋+1. (Remark: There is a formula, namely the nearest integer to ((n+3)^2)/(12), for the number of partitions of n into parts each of which is at most 3 but it is much more difficult to prove. There is also one for partitions with no part more than 4 , but it is even more complicated and difficult to prove.)
Uploaded: 2021-12-12T01:15:28-08:00
Duration: 4 min 43 s
Channel: Kevin Halasz

          Prove that the number of partitions of the positive integer $n$ into parts each of which is at most 2 equals $\lfloor n / 2\rfloor+1$. (Remark: There is a formula, namely the nearest integer to $\frac{(n+3)^{2}}{12}$, for the number of partitions of $n$ into parts each of which is at most 3 but it is much more difficult to prove. There is also one for partitions with no part more than 4 , but it is even more complicated and difficult to prove.)

Added by Esther J.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Kevin Halasz

12/12/2021

Step 1

Each partition can be uniquely identified by the sum of the 2s in the partition. ** Show more…

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Prove that the number of partitions of the positive integer $n$ into parts each of which is at most 2 equals $\lfloor n / 2\rfloor+1$. (Remark: There is a formula, namely the nearest integer to $\frac{(n+3)^{2}}{12}$, for the number of partitions of $n$ into parts each of which is at most 3 but it is much more difficult to prove. There is also one for partitions with no part more than 4 , but it is even more complicated and difficult to prove.)