All Questions
15
questions
0
votes
0
answers
48
views
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
votes
1
answer
54
views
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
0
votes
0
answers
12
views
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
views
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{...
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
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
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 $P$...
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
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
2
votes
1
answer
1k
views
Formula for how many combinations of powers of 2 sum to $2^n$
Given a number $2^n, n\in\mathbb{Z}\gt 0$, I would like to find a formula for how many unique sets of powers of $2$ sum to that number. This is related to the triangular numbers but excludes non-...
- 1,096
1
vote
1
answer
55
views
What is the name of the transform which finds the number of ways to make partitions of the given sizes?
I'm looking for the name of a transform which takes a sequence giving the number of 'prime' elements of a given size to the number of ways to make a number out of a sum of 'prime' elements, up to ...
- 32.3k
3
votes
2
answers
144
views
how interpret this partition identity?
use the symbol $P(N)$ to denote the set of all partitions of a positive integer $N$ and denote by $P_k$ the number of occurrences of $k$ in the partition $P \in P(N)$, so that
$$
N = \sum kP_k
$$
by ...
- 18.1k
1
vote
0
answers
47
views
What are all the possible sums (and how often do they occur) of a k-subsequence of an n-sequence of integers?
Let $A_n = \{a_1,\dots,a_n\}$ be a sequence of non-decreasing non-negative integers. Let $P(A_n,k)$ be the set of all subsequences of $A_n$ of length $k$. Given $n,k\in\mathbb Z_{\geqslant0}$ with $n\...
- 1,272
8
votes
3
answers
693
views
How to prove it? (one of the Rogers-Ramanujan identities)
Prove the following identity (one of the Rogers-Ramanujan identities) on formal power series by interpreting each side as a generating function for partitions:
$$1+\sum_{k\geq1}\frac{z^k}{(1-z)(1-z^2)...
- 825