All Questions
Tagged with integer-partitions sequences-and-series
45
questions
6
votes
1
answer
138
views
Prove that $\sum_{n=1}^{\infty}\frac{q^n}{1+q^{2n}}=\sum_{n=1}^{\infty}(d_1(n)-d_3(n))q^n$
Prove that $$\sum_{n=1}^{\infty}\frac{q^n}{1+q^{2n}}=\sum_{n=1}^{\infty}(d_1(n)-d_3(n))q^n$$ where $d_1(n)$ is the number of divisors of $n$ congruent to $1$ modulo $4$ and $d_3(n)$ is the number of ...
- 3,546
0
votes
1
answer
58
views
Number of all finite sequences of positive integers that add up to 100 (with some constraints)
Find the number of all finite sequences of integers $n_1, n_2, \ldots, n_k$, such that
$$
n_1 + n_2 + ⋯ + n_k = 100
$$
and such that for every $i \in \{1,\ldots,k\}$ we have $n_i \ge i$.
I have been ...
- 3
-1
votes
0
answers
2k
views
Find the number of good swaps.
Suppose we are given a positive integer N. Consider the sequence S=(1,2,…,N). You should choose two elements of this sequence and swap them.A swap is nice if there is an integer M (1≤M<N) such that ...
0
votes
1
answer
28
views
How does one handle series generating functions with multiple equals signs?
How would somebody walk through this equation? I'm looking for $q(n)$. If I'm given an input of 10, for example, how would this play out? The two equals signs is throwing me off. Does the result of ...
7
votes
2
answers
126
views
A sum of partitions
A friend of mine presented a problem I found interesting:
Compute the following: $$\sum_{n=0}^\infty\left(\prod_{k=1}^j(1+x^k)\right)[x^{mn}]$$ where $P(x)[x^n]$ denotes the $x^n$ coefficient of $P$...
2
votes
0
answers
192
views
Is there a generating function for this sequence?
The sequence is:
1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843,...
It is related to partitions of $n$. It is a ...
- 351
4
votes
0
answers
151
views
Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $
This identity is taken from a physics paper [1] stated without proof, on page 43.
$$ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty
=...
- 24.5k
0
votes
0
answers
181
views
A generating function $G(x)=-\frac{\frac{1}{x^5}(1+\frac{1}{x})(1-\frac{1}{x^2})}{((1-\frac{1}{x})(1-\frac{1}{x^3}))^2}$ related to partitions of $6n$
Fix a sequence $a_n={n+2\choose 2}$ of triangular numbers with the initial condition $a_0=1$,such that
$1,3,6,10,15,21,\dots$
given by
$F(x)=\frac{1}{(1-x)^3}=\sum_{n=0}^{\infty} a_n x^n\tag1$
...
- 2,813
1
vote
1
answer
92
views
Integer composition in exactly $T$ parts with maximum addend constraint.
In how many ways an integer $N$ can be partitioned into exactly $T$ parts such that
$T = \lfloor N/K \rfloor + 1$
$N = A_1 + A_2 + \cdots+ A_T$ where order matters
$0 \lt A_i \leq K$
$ N \bmod K \...
- 237
4
votes
1
answer
228
views
Prove or refute $\prod_{k=1}^\infty\left(\sum_{n=0}^\infty p(n)x^{kn}\right)^{\frac{\mu(k)}{k}}=e^{\frac{x}{1-x}}$
I believe that it is possible to combine the identity $(16)$ from this MathWorld related to the Möbius function, with the the generating function for the partition function $p(n)$, see in this ...
user243301
1
vote
0
answers
157
views
Ways to arrange in a two dimensional array an increasing sequence
Given a $n\times m$ grid, which has the number 1 in the upper-left square and a positive integer $1\leq k\leq n+m-1$ in the lower right-square, I am trying to determine in how many ways can the ...
- 11
2
votes
0
answers
719
views
$4\sum_{m,n=1}^{\infty}\frac{q^{n+m}}{(1+q^n)(1+q^m)}(z^{n-m}+z^{m-n})=8\sum_{m,n=1}^{\infty}\frac{q^{n+2m}}{(1+q^n)(1+q^{n+m})}(z^m+z^{-m})$
To prove the identity $$4\sum_{m,n=1}^{\infty}\frac{q^{n+m}}{(1+q^n)(1+q^m)}(z^{n-m}+z^{m-n})=8\sum_{m,n=1}^{\infty}\frac{q^{n+2m}}{(1+q^n)(1+q^{n+m})}(z^m+z^{-m})$$ I replaced $m-n$ by $k$ in LHS ...
2
votes
3
answers
2k
views
Algorithm for the number of partitions of $n$ into distinct parts
I am looking for an algorithm to find the number of ways of writing $n$ as a sum of positive integers without regard to order with the constraint that all integers in a given partition are distinct. ...
- 65
1
vote
1
answer
49
views
How we can partition $a_1, a_2, a_3, ... a_n$ in two sequence X and Y such that their sum of differences be minimum?
I couldn't write any algorithm that can do this in good order for $a_i < 100$ and $n < 2000$ :(
how we can partition $a_1, a_2, a_3, ... a_n$ in two sequence X and Y such that $|X_1 - X_2| + |...
1
vote
4
answers
1k
views
How many unordered partitions of $[n]$ are there, and how to enumerate?
In this context, a partition of $[n]$ is an assignment of each $1\le i\le n$ to a class, but in which the class names don't matter (can be given in any order). For example:
$[1]$ has one partition, $\...
- 27.7k