All Questions
Tagged with integer-partitions sequences-and-series
45
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48
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Find generating series on set of descending sequences, with weight function as taking sum of sequence
Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum ...
- 183
0
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1
answer
54
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Formula for number of monotonically decreasing sequences of non-negative integers of given length and sum?
What is a formula for number of monotonically decreasing sequences of non-negative integers of given length and sum? For instance, if length k=3 and sum n=5, then these are the 5 sequences that meet ...
- 15
1
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0
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60
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Generating Function for Modified Multinomial Coefficients
The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example,
$$\...
- 51
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12
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Prove convergence of partition sequence [duplicate]
We are given a partition of a positive integer $x$. Each step, we make a new partition of $x$ by decreasing each term in the partition by 1, removing all 0 terms, and adding a new term equal to the ...
5
votes
1
answer
238
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Prove that the general formula for a sequence $a_n$ is $\frac{(-1)^n}{n!}$
Here is a sequence $a_n$ where the first five $a_n$ are:
$a_1=-\frac{1}{1!}$
$a_2=-\frac{1}{2!}+\frac{1}{1!\times1!}$
$a_3=-\frac{1}{3!}+\frac{2}{2!\times1!}-\frac{1}{1!\times1!\times1!}$
$a_4=-\frac{...
1
vote
1
answer
57
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What is the 11th unordered combination of natural numbers that add upto 6 in the partition function?
So, I was making unordered combinations of natural numbers which add upto a certain natural number. I was able to go till 6 when I got to know about the partition function. I was pleased to see that ...
- 177
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127
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An identity related to the series $\sum_{n\geq 0}p(5n+4)x^n$ in Ramanujan's lost notebook
While browsing through Ramanujan's original manuscript titled "The Lost Notebook" (the link is a PDF file with 379 scanned pages, so instead of a click it is preferable to download) I found ...
- 88.3k
1
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1
answer
143
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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
2
votes
2
answers
81
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Show that series converges by estimating number of partitions into distinct parts
I need some help with solving the following problem: Let $Q(n)$ be the number of partitions of $n$ into distinct parts. Show that $$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$ is convergent by estimating $Q(n)...
- 155
7
votes
1
answer
167
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How to prove the following resummation identity for Erdős–Borwein constant?
Question:
How to prove
$$\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}$$
for all $q \in \mathbb{C}$ such that $\left|q\right| < 1$?
...
- 3,132
0
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0
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34
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How might I go about (dis)proving a conjecture involving RMS, integer compositions, and contiguous subsequences of finite sequences?
I'm here for some help to prove or disprove a (possibly trivial) conjecture concerning compositions (i.e. ordered partitions) of the natural number $n$, and the contiguous subsequences that they ...
1
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2
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193
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What is the closed form solution to the sum of inverse products of parts in all compositions of n?
My question is exactly as in the title:
What is the generating function or closed form solution to the sum of inverse products of parts in all compositions of $n$?
This question was inspired by just ...
- 171
2
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1
answer
38
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Sum of $k-tuple$ partition of $d$. $k$ and $d$ are both positive integers.
I am trying to understand this paper by Bondy and Jackson and in it I found the following calculation in which I cannot figure out how we got from 2nd last step to last step. I have referred the paper ...
5
votes
1
answer
371
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What's the sequence $3,9,24,21,36,30,75,120,270,462,837,1320,2085,\ldots$?
In order to find a formula of the partition of an integer into 5 parts (see $(6)$ below), I find the sequence
$$S_1: 3,9,24,21,36,30,75,120,270,462,837,1320,2085,\ldots \tag1$$
It's clear that $S_1$ ...
5
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3
answers
1k
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Proving that odd partitions and distinct partitions are equal
I am working through The Theory of Partitions by George Andrews (I have the first paperback edition, published in 1998).
Corollary 1.2 is a standard result that shows that the number of partitions of $...
6
votes
1
answer
138
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Prove that $\sum_{n=1}^{\infty}\frac{q^n}{1+q^{2n}}=\sum_{n=1}^{\infty}(d_1(n)-d_3(n))q^n$
Prove that $$\sum_{n=1}^{\infty}\frac{q^n}{1+q^{2n}}=\sum_{n=1}^{\infty}(d_1(n)-d_3(n))q^n$$ where $d_1(n)$ is the number of divisors of $n$ congruent to $1$ modulo $4$ and $d_3(n)$ is the number of ...
- 3,546
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1
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58
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Number of all finite sequences of positive integers that add up to 100 (with some constraints)
Find the number of all finite sequences of integers $n_1, n_2, \ldots, n_k$, such that
$$
n_1 + n_2 + ⋯ + n_k = 100
$$
and such that for every $i \in \{1,\ldots,k\}$ we have $n_i \ge i$.
I have been ...
- 3
-1
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0
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2k
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Find the number of good swaps.
Suppose we are given a positive integer N. Consider the sequence S=(1,2,…,N). You should choose two elements of this sequence and swap them.A swap is nice if there is an integer M (1≤M<N) such that ...
0
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1
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28
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How does one handle series generating functions with multiple equals signs?
How would somebody walk through this equation? I'm looking for $q(n)$. If I'm given an input of 10, for example, how would this play out? The two equals signs is throwing me off. Does the result of ...
7
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2
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126
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A sum of partitions
A friend of mine presented a problem I found interesting:
Compute the following: $$\sum_{n=0}^\infty\left(\prod_{k=1}^j(1+x^k)\right)[x^{mn}]$$ where $P(x)[x^n]$ denotes the $x^n$ coefficient of $P$...
2
votes
0
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192
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Is there a generating function for this sequence?
The sequence is:
1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843,...
It is related to partitions of $n$. It is a ...
- 351
4
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151
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Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $
This identity is taken from a physics paper [1] stated without proof, on page 43.
$$ \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
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181
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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
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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
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0
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157
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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
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719
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$4\sum_{m,n=1}^{\infty}\frac{q^{n+m}}{(1+q^n)(1+q^m)}(z^{n-m}+z^{m-n})=8\sum_{m,n=1}^{\infty}\frac{q^{n+2m}}{(1+q^n)(1+q^{n+m})}(z^m+z^{-m})$
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
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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
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1
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49
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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
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4
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1k
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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, $\...
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