Let q(n) be the number of partitions of n whose largest summand occurs exactly once. Show that q(n) = p(n - 1) for all n >= 1.
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For all integers n > 2, show that the number of integer partitions of n in which each part is greater than one is given by p(n)-p(n-1), where p(n) is the number of integer partitions of n.
Sri K.
a) Show that the number p(n) of unordered integer partitions of an integer n equals the number pn(2n) of unordered integer partitions of length n of the integer 2n.
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.)
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