Find a formula for the number of partitions of n into parts of size at most 4.
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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 ⌊ 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.)
Krishna G.
Text: Calculate the number of partitions p(n) of a finite set of n elements from n=1 to n=50. Or you can do it by hand without a computer or describe how to do it by hand. Examples: n=2, p(2)=2 {a,b} {a},{b} n=3, p(3)=5 {a,b,c} {ab},{c} {ac},{b} {bc},{a} {a},{b},{c} n=4, p(4)=15 {abcd}, 1 {abc},{d}, 2 {acd},{b}, 3 {abd},{c}, 4 {bcd},{a}, 5 {ab},{cd}, 6 {ac},{bd}, 7 {ad},{bc}, 8 {ab},{c},{d}, 9 {ac},{b},{d}, 10 {ad},{b},{c}, 11 {bc},{a},{d}, 12 {cd},{a},{b}, 13 {bd},{a},{c}, 14 {a},{b},{c},{d}, 15
Shannon K.
Find the generating function for the number of partitions of an integer n, where you can use the integer k at most k times in the partition. As an example, the following at the allowable partitions of 5 that we are looking at here: 5 good 4 + 1 good 3 + 2 good 3 + 1 + 1 not good 2 + 2 + 1 not good 2 + 1 + 1 + 1 not good 1 + 1 + 1 + 1 + 1 not good Not a lot of calculation or work is needed to answer this question, so we will grade you mostly on your explanation.
Sri K.
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