All Questions
Tagged with integer-partitions generating-functions
210
questions
4
votes
1
answer
134
views
The total number of all different integers in all partitions of n with smallest part $\geq 2$
I want to show that the total number of all different integers in all partitions of $n$ with smallest part $ \geq 2$ is $p(n-2)$.
Example: partitions of $7$ with smallest part $ \geq 2$ are (7), (5,2),...
- 53
2
votes
2
answers
168
views
Finding the generating function.
Find the generating function for the number of solutions for the equation $x_1+x_2+x_3+x_4 = n$, where $x_1,x_2,x_3,x_4\geq1$, and $x_1 < x_2$.
My attempt so far: I have tried putting a $y$ value ...
- 165
1
vote
1
answer
143
views
Generating function of ordered odd partitions of $n$.
Let the number of ordered partitions of $n$ with odd parts be $f(n)$. Find the generating function $f(n)$ .
My try : For $n=1$ we have $f(1)=1$, for $n=2$, $f(2)=1$, for $n=3$, $f(3)=2$, for $n=4$, $...
user1137804
1
vote
1
answer
92
views
Identity about generating function related to binary expression of integers
For any nonnegative integer $n$, let $\mu(n)$ be $1$ if the binary expression of $n$ contains even number of ones; and $-1$ if the binary expression of $n$ contains odd number of ones. For example, $\...
user1089451
1
vote
1
answer
96
views
Representation of number as a sums and differences of natural numbers
Lets consider all the combinations of:
$$1+2+3+4=10,\ \ 1+2+3-4=2,\ \ 1+2-3+4=4,\ \ 1+2-3-4=-4, $$
$$1-2+3+4=6,\ \ 1-2+3-4=-2,\ \ 1-2-3+4=0,\ \ 1-2-3-4=-8,$$
$$-1+2+3+4=8,\ \ -1+2+3-4=0,\ \ -1+2-3+4=2,...
- 1,818
1
vote
0
answers
49
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Generating function of sequence related to binary expression of integer as follows
For any $n \in \Bbb N$, let $O(n)$ be $1$ if the binary expression of $n$ contains even number of ones; and $-1$ if the binary expression of $n$ contains odd number of ones, and $Z(n)$ be $1$ if the ...
user1089451
0
votes
0
answers
178
views
Generating function of integer partitions at most m parts
The generating function of partition integer with the maximum number is m is $$
\frac{x^m}{(1-x)(1-x^2)\cdots(1-x^m)}
$$
It's quite easy to understand if you expand it $$
(1+x+x^2+\cdots)(1+x^2+x^4+\...
2
votes
1
answer
178
views
two-variable generating function for all number partitions
I'm stuck to solve the problems below:
(a) Let $p(n, k)$ be the number of partitions of $n$ into exactly $k$ parts. Show that
$$
\sum_{n, k \geq 0} p(n, k) x^{n} t^{k}=\prod_{i \geq 1} \frac{1}{1-x^{i}...
- 143
1
vote
3
answers
73
views
Partitions without repetition
I want to know how many partitions without repetition 19 has. I know I should see the coefficient of $x^{19}$ in $$\prod_{k=1}^\infty(1-x^k),$$
but i'm having trouble finding it. Ay hint?
- 153
1
vote
1
answer
86
views
Number of partitions with distinct even parts/parts with multiplicity $\leq 3$
I am supposed to solve a problem regarding partitions of $n \in \mathbb{N}$ into:
distinct even parts
parts with multiplicity $\leq 3$
I am supposed to prove that 1. and 2. are equal.
So I tried ...
- 333
7
votes
1
answer
167
views
How to prove the following resummation identity for Erdős–Borwein constant?
Question:
How to prove
$$\sum_{m=1}^{\infty}\left(1-\prod_{j=m}^{\infty}(1-q^j)\right) = \sum_{n=1}^{\infty}\frac{q^n}{1-q^n} \tag{1}$$
for all $q \in \mathbb{C}$ such that $\left|q\right| < 1$?
...
- 3,132
1
vote
0
answers
49
views
Coefficients in Gaussian polynomials [duplicate]
Let
$$
\binom{n}{k}_{\!q} = \frac{(1-q^n) \cdots (1-q^{n-k+1})}{(1-q) \cdots (1-q^k)}
$$
be the Gaussian polynomials. For example,
$$
\binom{6}{3}_{\!q} = 1+q+2q^2+3q^3+3q^4+3q^5+3q^6+2q^7+q^8+q^9.
$$
...
- 175
-3
votes
4
answers
129
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Find a minimal set whose elements determine explicitly all integer solutions to $x + y + z = 2n$
Is there a way to exactly parameterise all the solutions to the equation $x + y + z = 2n$, for $z$ less than or equal to $y$, less than or equal to $x$, for positive integers $x,y,z$?
For example, for ...
- 67
0
votes
1
answer
138
views
Infinite product expression of partition function
I'm working on a problem (specifically, I'm using an exam paper without course notes to prepare for a course starting in September),
Define the partition function $P(q)$ and give its infinite product ...
- 2,281
1
vote
1
answer
74
views
Counting Partitions of $k$ into $j$ distinct parts, with sizes restricted by a sequence.
This question has come up in my research, but since I usually do not do combinatorics, I am struggling to find out any information regarding these sequences of numbers.
Let $L_n = L_1, L_2\dots$ be a ...
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