All Questions
Tagged with integer-partitions generating-functions
210
questions
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3
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80
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Proving that $t_n = r_n$
Given a non-negative integer $n,$ let $t_n$ be the number of ways to partition $n$ using only powers of $2$ where each power is used at most three times, and let $r_n$ be the number of ways to ...
- 1,150
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0
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55
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Number of integer partitions of $n$ with exactly $k$ parts from the set $S = \{s_1, s_2, \ldots, s_m\}$.
Specifically, I want to find the number of integer partitions of 1000 into exactly 100 parts of sizes {5, 20, 100} ignoring different orderings. I am trying to find the generating function. I have ...
2
votes
1
answer
144
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An identity involving partitions
I have been trying to prove an identity using a combinatorial argument or another technique. I need to define the following first:
$$R(n,m) = \{ (r_1,r_2,...,r_n): \sum_{i=1}^n r_i = m \text{ and } ...
1
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0
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43
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Systematic vertex information about finite Young's lattice
Suppose that I have Young's lattice $Y_{\mu}$ (finite sublattice for everything included in $\mu$). See the red area in the linked picture I found online where $\mu=[22]$ and note that the arrows ...
- 21
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56
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Finding Generating Function of Series, Coefficients Relate to Partitions
Let $\displaystyle p_{\leq c}(n)$ represent the number of partitions of $n$ into at most $c$ parts. What is the generating function of $\displaystyle\sum_{n \geq 0} p_{\leq c}(n)x^n$?
I'm completely ...
7
votes
3
answers
234
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Prove that $\prod_{i\geq 1}\frac{1}{1-xy^{2i-1}} = \sum_{n\geq 0} \frac{(xy)^{n}}{\prod_{i=1}^{n}\left( 1-y^{2i} \right)}.$
Prove that
$$\prod_{i\geq 1}\frac{1}{1-xy^{2i-1}} = \sum_{n\geq 0} \frac{(xy)^{n}}{\prod_{i=1}^{n}\left( 1-y^{2i} \right)}.$$
Here I am trying the following
\begin{align*}
\prod_{i\geq 1}\frac{1}{1-xy^...
- 376
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1k
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Proving that odd partitions and distinct partitions are equal
I am working through The Theory of Partitions by George Andrews (I have the first paperback edition, published in 1998).
Corollary 1.2 is a standard result that shows that the number of partitions of $...
3
votes
1
answer
56
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The number of binary sequences generated by a parity map from all integer partitions of $n$ and their permutations
Let $n$ be an integer and $n=(n_1,\dots,n_n)$ all its partitions. The partitions with less than $n$ integers (all but one) are padded by zeros from the left AND sorted such that first come even ...
- 33
1
vote
1
answer
819
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Partitions of $n$ into even number of parts versus into odd number of parts
I'm under the impression that if $n$ is even then the number of partitions of $n$ into an even number of parts exceeds the number of partitions of $n$ into an odd number of parts. And the opposite if $...
- 462
2
votes
2
answers
128
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Counting the number of partitions such that every part is divisible by $k$.
I would like to do this using generating functions. I'm comfortable with the generating function for the number of partitions of $n$: $$(1+x+x^2+\dots)(1+x^2+x^4+\dots)(1+x^3+x^6+\dots)\cdot\dots,$$ ...
- 462
3
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answers
104
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Determine in how many ways the amount of 50 can be paid by bank notes of 1, 2, 5, 10, and 20
Determine in how many ways the amount of 50 can be paid by bank notes of 1, 2, 5, 10, and 20
I have used the generating function that
$G(x)=\frac{1}{1-x}\frac{1}{1-x^2}\frac{1}{1-x^5}\frac{1}{1-x^{10}...
- 63
0
votes
3
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256
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How many ways can you pay $100 when the order in which you pay the coins matters?
Suppose there is a vending machine where you have to pay $100. You
have an unlimited amount of 1 dollar, 2 dollar and 5 dollar coins. How many ways
can you pay when the order in which you pay the ...
1
vote
1
answer
78
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Computing problems and generating functions
Q. Find the generating function for the sequence $\{a_n\}$, where $a_n$ is the number of solutions to the equation: $a+b+c=k$ when $a, b, c$ are non-negative integers such that $a\ge2, 0\le b\le3$ and ...
0
votes
1
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44
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On the partitions of a number
I need to find the generating function for number of partitions of $n$ into parts that are divisible by $2$ and every one of them appears at most twice.
I think that solution is $f(x)=(1+x+x^2)(1+x^2+...
- 533
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1
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109
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Find number of solutions for equation: $~x+y+z=n~$ where $~x,~y,~z~$ are non-negative whole numbers and $~x\le y\le z~$.
Find number of solutions for equation: $~x+y+z=n~$ where $~x,~y,~z~$ are non-negative whole numbers and $~x\le y\le z~$.
First I used substitution $~y=x+k,~ z=y+k~$ where $~k\ge 0~$(that is $y=x+k, z=...
- 533