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.

Expert Solution

Colby Messinger answered 2 years ago

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.

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.

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

Let the cardinal number of N, the set of all natural numbers, be א0. Prove that...

Let the cardinal number of N, the set of all natural numbers, be א0. Prove that the product set N × N = {(m,n);m ∈ N,n ∈ N} has the same cardinal number. Further prove that Q+, the set of all positive rational numbers, has the cardinal number N_0. Hint: You may use the formula 2^(m−1)(2n − 1) to define a function from N × N to N, see the third example on page 214 of the textbook.

Let N be the number of requests to a web server per day and let N...

Let N be the number of requests to a web server per day and let N Poisson(). Each request comes from a human with probability p or from a spam bot with probability 1 ? p. Assume that the requests are independent of each other. Let X be the number of requests from humans per day and Y be the number of requests from spam bots per day. (a) State the conditional distribution of X given N = n, and...

Recognizing partitions - sets of strings. (b) Let A be the set of words in the...

Recognizing partitions - sets of strings. (b) Let A be the set of words in the Oxford English Dictionary (OED). For each positive integer j, define Aj to be the set of all words with j letters in the OED. For example, the word "discrete" is an element of A8 because the word "discrete" has 8 letters. The longest word in the OED is "pneumonoultramicroscopicsilicovolcanoconiosis" which has 45 letters. You can assume that for any integer i in the range...

In the binomial probability distribution, let the number of trials be n = 4, and let...

In the binomial probability distribution, let the number of trials be n = 4, and let the probability of success be p = 0.3310. Use a calculator to compute the following. (a) The probability of three successes. (Round your answer to three decimal places.) (b) The probability of four successes. (Round your answer to three decimal places.) (c) The probability of three or four successes. (Round your answer to three decimal places.)

c) Let R be any ring and let ??(?) be the set of all n by...

c) Let R be any ring and let ??(?) be the set of all n by n matrices. Show that ??(?) is a ring with identity under standard rules for adding and multiplying matrices. Under what conditions is ??(?) commutative?

Let S(n) be the number of subsets of {1,2,...,n} having the following property: there are no...

Let S(n) be the number of subsets of {1,2,...,n} having the following property: there are no three elements in the subset that are consecutive integers. Find a recurrence for S(n) and explain in words why S(n) satisfies this recurrence

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum...

Let τ (n) denote the number of positive divisors of n and σ(n) denote the sum of the positive divisors of n (as in the notes). (a) Evaluate τ (1500) and σ(8!). (b) Verify that τ (n) = τ (n + 1) = τ (n + 2) = τ (n + 3) holds for n = 3655 and 4503. (c) When n = 14, n = 206 and n = 957, show that σ(n) = σ(n + 1).

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