Prove that the number of partitions of n into parts of size 1 and 2 is...

Prove that the number of partitions of n into parts of size 1 and 2 is equal to the number of partitions of n + 3 into exactly two distinct parts

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Colby Messinger answered 2 years ago

Partitions Show that the number of partitions of an integer n into summands of even size...

Partitions Show that the number of partitions of an integer n into summands of even size is equal to the number of partitions into summands such that each summand occurs an even number of times.

For all integers n > 2, show that the number of integer partitions of n in...

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.

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is...

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is equal to the number of partitions of n in which the two largest parts are equal.

Let N(n) be the number of all partitions of [n] with no singleton blocks. And let...

Let N(n) be the number of all partitions of [n] with no singleton blocks. And let A(n) be the number of all partitions of [n] with at least one singleton block. Prove that for all n ≥ 1, N(n+1) = A(n). Hint: try to give (even an informal) bijective argument.

If n>=2, prove the number of prime factors of n is less than 2ln n.

Prove by induction on n that the number of distinct handshakes between n ≥ 2 people...

Prove by induction on n that the number of distinct handshakes between n ≥ 2 people in a room is n*(n − 1)/2 . Remember to state the inductive hypothesis!

Using the pumping lemma, prove that the language {1^n | n is a prime number} is...

Using the pumping lemma, prove that the language {1^n | n is a prime number} is not regular.

prove 2 is a factor of (n+1)(n+2) for all positive integers

Combinatorics Prove bell number B(n)<n!

*Combinatorics* Prove bell number B(n)<n!

Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1...

Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1 distinct degrees (up to isomorphism). The other answers on the website are incorrect.

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