All Questions
20 questions
3
votes
1
answer
439
views
An identity for polynomials over partitions
Given an integer partition $\lambda=(\lambda_1,\dots,\lambda_{\ell(\lambda)})$ of $n$ where $\ell(\lambda)$ is the length of $\lambda$, associate its conjugate partition $\lambda'$. Denote by $\lambda'...
- 43.2k
26
votes
2
answers
1k
views
Partitions to different parts not exceeding $n$
Consider the polynomial $(1+x)(1+x^2)\dots (1+x^n)=1+x+\dots+x^{n(n+1)/2}$, which enumerates subj. How to prove that it's coefficients increase up to $x^{n(n+1)/4}$ (and hence decrease after this)? Or ...
- 109k
11
votes
3
answers
1k
views
A problem on a specific integer partition
Let $n$ be a positive integer, we consider partitions of the following form :
$$n = d^{2}_{1} + d^{2}_{2} + ... + d^{2}_{r}$$ such that :
$d_{i}\vert n$
$1=d_{1}<d_{2} \le d_{3} \le ... \le d_{r}$...
9
votes
2
answers
388
views
Every possible number of partitions by restricting parts?
Write $p(n)$ for the number of integer partitions of $n$. For $S \subseteq \{1, \ldots, n\}$, let $p_S(n)$ be the number of partitions of $n$ with all parts in $S$. So $p(n) = p_{\{1,\ldots,n\}}(n)$....
- 4,589
15
votes
2
answers
1k
views
hook-length formula: "Fibonaccized" Part I
Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \lambda$, define the hook numbers $h_{(i,j)} = \lambda_i + \lambda_j' -i - j +1$ where $\...
- 43.2k
13
votes
2
answers
803
views
Two interpretations of a sequence: an opportunity for combinatorics
The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function
$$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$
In particular, look ...
- 43.2k
12
votes
3
answers
892
views
Set partitions and permanents
Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd.
...
- 884
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
10
votes
4
answers
1k
views
Binomial coefficient in Andrews' partition book
First of all, I think MathOverflow is a very great community to discuss math, either basic or advanced, and I'm glad to participate here. It's my first post, so I'm sorry if i did anything wrong, and ...
- 103
8
votes
1
answer
472
views
In search of a combinatorial reasoning for a vanishing sum
Assume $s, j \in\mathbb{N}$. Define the set
$$\mathcal{A}_{j,s}:=\{(n_1,n_2,\dots,n_j)\in\mathbb{Z}_{\geq0}^j\vert \,
n_1+2n_2+\cdots+jn_j=j, \, n_1+n_2+\cdots+n_j=s\}.$$
Question. Is there a ...
- 43.2k
8
votes
1
answer
368
views
generalizing Wilf's conjecture: Uppuluri-Carpenter numbers
The complementary Bell numbers have the exponential generating function
$$\sum_{n\geq0}\tilde{B}_nx^n=e^{1-e^x}.$$
Herb Wilf conjectured that $\tilde{B}_n=0$ only for $n=2$. By now, there are a few ...
- 43.2k
6
votes
4
answers
1k
views
Counting refinements of partitions
Let $p$ and $q$ be partitions of $n$. We say $q$ refines $p$ if the parts of $p$ can be subdivided to produce the parts of $q$. For example, $(5,5,1)$ refines $(6,5)$ but not $(7,4)$. $(n)$ refines ...
- 1,226
6
votes
1
answer
392
views
hook-length formula: "Fibonaccized": Part II
This is a natural follow-up to my previous MO question, which I share with Brian Hopkins.
Consider the Young diagram of a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$. For a square $(i,j) \in \...
- 43.2k
6
votes
1
answer
305
views
Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative?
In this paper, the following result is proved.
For any prime $p$, all the Fourier coefficients of
$$\eta(q^p)^p / \eta(q) = q^{\frac{p^2-1}{12}} \prod_{n=1}^\infty (1 - q^{pn})^p (1 - q^{n})^{-1}$$
...
- 427
5
votes
1
answer
204
views
Collapsed partitions and generating functions
Given $n\in\Bbb{N}$, the number of (unrestricted) integer partitions of $n$ are given by
$$\sum_{n\geq0}p(n)x^n=\prod_{j\geq1}\frac1{1-x^j}.$$
Define the collapsed partitions of $n$ to be the ...
- 43.2k
4
votes
2
answers
395
views
Congruence for complementary Bell numbers
The Bell numbers $B(n)$ can be given as a sum of the (signed) Stirling numbers of the second kind $S(n,k)$ as $B(n)=\sum_{k=0}^nS(n,k)$. There are also the so-called complementary Bell numbers defined ...
- 43.2k
3
votes
1
answer
251
views
Congruence modulo 2 for q-series
This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks.
I would like to ask:
QUESTION. Is this congruence true ...
- 43.2k
2
votes
1
answer
2k
views
Number of 3-tuple partitions of a multiple of three which follow the triangle inequality
Given n=3t, t$\in \mathbb N$; let $\mathbb L_3$ be set of all distinct integer partitions of n having 3 parts; say $\lambda_1,\lambda_2,\lambda_3$ .
If I chose any one partition randomly from $\...
- 851
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
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