All Questions
28 questions with no upvoted or accepted answers
15
votes
0
answers
767
views
Wherefore art thou a Borcherds Product?
This question essentially asks how can one recognize (or rule out) that a generating function of combinatorial origin may be given as a Borcherds type product. I'll start with a motivational example: ...
- 85.6k
12
votes
0
answers
643
views
Wilf's conjecture: complementary Bell numbers
The complementary Bell numbers or Uppuluri–Carpenter numbers, denoted $\tilde{B}_n$, can be delivered by
$$G(x):=\sum_{n\geq0}\tilde{B}_n\frac{x^n}{n!}=e^{1-e^x}.$$
Definition. Fix an integer $m\geq0$....
- 43.2k
11
votes
0
answers
290
views
Color your partitions by parity
Let $a_c(n)$ be the number of ways to partition a positive integer $n$ where each even part comes in $c$ colors. Then, we can supply the generating function
$$\sum_{n\geq0}a_c(n)q^n=\prod_{k\geq1}\...
- 43.2k
7
votes
0
answers
251
views
Can the partition function $p(n)$ take perfect power values?
Recall that the perfect powers are those integers $m^k$ with $k,m\in\{2,3,\ldots\}$. I don't consider $0$ or $1$ as a perfect power.
Y. Bugeaud, M. Mignotte and S. Siksek [Annals of Math., 2006] ...
- 15.6k
6
votes
0
answers
171
views
An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
5
votes
0
answers
140
views
Complementary Bell numbers $B^{\pm}(24n+14)$
The complementary Bell numbers $B^{\pm}(n)$ are defined by the alternating sum of the Stirling numbers of the second kind, $S(n,k)$:
$$B^{\pm}(n)=\sum_{k=0}^n(-1)^kS(n,k),$$ and they count the ...
- 407
5
votes
0
answers
194
views
Lemmas involving two partitions of integers
Question: Does anyone know a reference to the following lemmas involving two partitions? (The proofs are not hard, and may well be previously recorded, but where?) First some notation. Let $r$ be a ...
- 1,991
4
votes
0
answers
211
views
Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions
Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
- 261
4
votes
0
answers
67
views
$3$-variable Jacobi style identity linked to generalised Frobenius partitions
I was fiddling around with a family of probabilistic models and came across two "identities", which appear to be linked to generalized Frobenius partitions (more on this below). I would be ...
- 562
3
votes
0
answers
213
views
A family of polynomials related to integer partitions
For a positive integer $n$, let $p(n)$ be the number of partitions of $n$.
For $1\le k\le n$, let $p(n,k)$ denote the number of partitions of $n$ having exactly $k$ terms; in other words, $p(n,k)$ is ...
- 15.6k
3
votes
0
answers
120
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,928
3
votes
0
answers
254
views
Enumerating multi-core binary partitions
An integer partition $\lambda$ of $n$ is called a binary partition provided that its parts are powers of $2$ (dyadic). Example: Let $n=3$. The binary partitions are $\lambda=(2,1)$ and $\lambda=(1,1)$ ...
- 43.2k
3
votes
0
answers
258
views
Partitions of n into k distinct parts which are multiples of given numbers
Is there anything known about the number of partitions of an integer $n$ into $k$ distinct parts in the following way?
Let $a_1,\dotsc,a_k\geqslant1$ be given integers. In how many ways can we write $...
- 539
3
votes
0
answers
110
views
Integer partitions under divisibility constraint
Consider integer partitions of $x \in \mathbb{N}$ of size $k$ under the constraint that the partition elements are distinct and the ratio of any element to each smaller element is a natural number.
...
- 2,558
2
votes
0
answers
72
views
Recursion for the number of partitions of $m^n-1$ into powers of $m$
Let $a(n,m)$ be the number of partitions of $m^n-1$ into powers of $m$. In other words,
$$a(n,m)=[z^{m^n-1}] \prod\limits_{k\geqslant 0} \frac{1}{1-z^{m^k}}$$
Let
$$
R(n,m,q)=\sum\limits_{j=0}^{m(q+1)-...
- 4,928
2
votes
0
answers
228
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 43.2k
2
votes
0
answers
135
views
Writing integers as sequences of products by 2 and integer divisions by 3
For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$.
For instance:
$$
100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
- 1,444
2
votes
0
answers
123
views
Coefficents of these partition-based polyomials are $0, \pm1$
This is a follow up on my earlier MO question.
Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate $\...
- 43.2k
2
votes
0
answers
140
views
Number of multipartite partitions with odd components
For some positive integer $r$, by an $r$-vector I will mean an $r$-tuple $(a_1,a_2,\dots,a_r)$ with $a_1,\dots,a_r$ nonnegative integers not all zero, and I will call it odd if $a_1,\dots,a_r$ are all ...
- 767
1
vote
0
answers
95
views
Pretty simple recursion for the A290383
Let $a(n)$ be A290383 i.e. number of set partitions of $[n]$ such that the smallest element of each block is odd. Here
$$
a(n)=b(n,0,0)
$$
where
$$
b(n,m,t)=\sum\limits_{j=1}^{m-t+1}b(n-1,\max(m,j),1-...
- 4,928
1
vote
0
answers
170
views
A representation problem involving strict partition numbers
For each positive integer $n$, let $q(n)$ denote the number of ways to write $n$ as a sum of distinct positive integers. We call those $q(n)\ (n=1,2,3,\ldots)$ strict partition numbers.
The sequence $...
- 15.6k
1
vote
0
answers
100
views
Conjecture on numbers $k$ having only one partition into parts with same binary weight as a binary weight of $k$
Let $\operatorname{tr}(n)$ be A007814, number of trailing zeros in the binary representation of $n$.
Also, let $\operatorname{ntr}(n)$ be A086784, number of non-trailing zeros in the binary ...
- 4,928
1
vote
0
answers
156
views
Number-theoretic proof of integrality of a fraction and asymptotics of sum over partitions related to symmetric group
Consider $\;\alpha=(\alpha_1,...,\alpha_n)\in\mathbb{Z}_+^n\;$ such that $\;1\alpha_1+...+n\alpha_n=n.\;$ Let $\varphi$ denote Euler totient-function.
Let $\;T_\alpha\;$ be a set of permutations in $...
- 651
1
vote
0
answers
203
views
Generalizing "partition into odd parts=partition into distinct parts"?
The number of partitions into distinct parts is known to agree with the number of partitions with odd parts. For instance, this follows from
$$\prod_{k=1}^{\infty}(1+q^k)=\prod_{n=1}^{\infty}\frac1{1-...
- 43.2k
1
vote
0
answers
207
views
Parity of number of partitions of $n!/6$ and $n!/2$
The parities of the number of partitions of $n!/6$ and $n!/2$ appear to be non-random initially, as follows — is there an explanation for this other than chance? With $p$ being the partition ...
- 105
1
vote
0
answers
105
views
Factorial Sums over Compositions or ``Unlabeled Permutations"
Let $C_n$ denote subset of integer compositions of $n$ and $c=(c_1,c_2,\dots c_n)$
In a divergent sum, the sequence
$$
a_n=\sum_{c\in C_n} \prod_{c_i\in c} c_i!
$$
frequently shows up and one ...
- 1,306
0
votes
0
answers
186
views
Are the numbers $\sum_{n=1}^\infty\frac1{p(n)}$ and $\sum_{n=1}^\infty\frac1{q(n)}$ transcendental?
For each positive integer $n$, let $p(n)$ be the number of partitions of $n$ (i.e., the number of ways to write $n$ as a sum of positive integers), and let $q(n)$ be the number of strict partitions of ...
- 15.6k
0
votes
0
answers
80
views
Minimizing coefficients in a product related to the Rogers Ramanujan identity
Start with the product for partitions into parts congruent to $1$ or $4$ modulo $5$:
$(1 + x + x^2 + x^3 + ...)(1 + x^4 + x^8 + x^{12} +...)(1 + x^6 + x^{12} + x^{18} +...)$...
Now replace some of the ...
- 211