0
votes
2answers
28 views

Alternative reference for number of restricted partitions

I am looking for the number of partitions of some number $n$ into $k$ parts. Following the Wikipedia article on partitions, I ended up with Andrew's book [1]. Judging by Google's preview Chapter 3 ...
0
votes
1answer
15 views

Regular expression for a particular language

Several years ago I came across a paper that defined a regular expression (or collection of regular expressions?) for a specific language. The language is the language of set partitions enumerated by ...
1
vote
0answers
47 views

Monoid on ordered partitions of a natural number

Fix a natural number $n$, and let $O_n$ be the set of ordered partitions of $n$. For example $O_3=\{1+1+1,1+2,2+1,3\}$ which can also be written as $\{1|2|3,1|23,12|3,123\}$. We can define two ...
2
votes
0answers
30 views

Partitions and divisor functions: what is known about their relations?

If $i\geq 1$ is an integer, we have the following integer valued functions (for any integer $n\geq 0$): \begin{align} p_i(n)&=\textrm{the number of }i\textrm{-dimensional partitions of ...
4
votes
1answer
122 views

Formula for evaluation of character on a transposition

Let $\lambda\vdash n$ be a partition of $n\in\mathbb N$ and $\chi=\chi_\lambda$ the corresponding irreducible character of the symmetric group $S_n$. Denote by $\lambda^t$ be the transpose of ...
2
votes
0answers
89 views

What is the correct technical term for this generalization of an integer partition?

Given a vector $v$ with non-negative integer coordinates, is there a technical term for an unordered tuple of vectors $(v_1,\dots, v_k)$ with non-negative integer coordinates such that $v_1+\dots+v_k ...
4
votes
6answers
412 views

Book recommendation for Integer partitions and q series

I have been studying number theory for a little while now, and I would like to learn about integer partitions and q series, but I have never studied anything in the field of combinatorics, so are ...
2
votes
1answer
509 views

Partitions of an integer into k parts.

I am interested in knowing whether an exact formula (analogous to the Hardy-Ramanujan-Rademacher formula for $p(n)$) for the number of partitions of a positive integer into k parts is known. I tried ...
1
vote
0answers
101 views

Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
0
votes
1answer
35 views

The “theory of compound partitions”

I was reading a sort of mini-bio on Sylvester the other day and a "Theory of Compound Partitions" was mentioned in the discussion of his research interests. I wanted to ask, is this the same or the ...
8
votes
2answers
344 views

Closed-form Expression of the Partition Function $p(n)$

I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ ...
7
votes
1answer
196 views

Seeking a textbook proof of a formula for the number of set partitions whose parts induce a given integer partition

Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq ...
1
vote
1answer
50 views

Notation for “duplicating” partitions

I'm using Macdonald's "Symmetric Functions and Hall Polynomials" as a reference and did not find what I was looking for -- apologies if I only missed it. As an example, let us consider the partition ...