3
votes
0answers
43 views

Asymptotics of partitions in at most n parts, bounded by r

For every positive integers $n,r,w$ define $$ p_w(n,r)=\#\{ (i_1,...,i_r) | \, 0\leq i_1 \leq \dots \leq i_r\leq n, \, i_1+\dots+i_r=w\} $$ as the number of partitions of $w$ in at most $r$ piece ...
6
votes
0answers
49 views

Partitioning points with a line

Let $A_{m, n} = \{1, 2, \dots, n\} \times \{1, 2, \dots, m\}$. A straight line would partition the points into two sets. How many ways are there to do it? Let $p_{m, n}$ be that number. Apparently ...
1
vote
1answer
62 views

Help understanding solution to growth of partition function

I'm currently a Combinatorics student trying to parse through this solution. I do not understand the proof currently. Any help understanding it is greatly appreciated. Question Let the number of ...
0
votes
0answers
39 views

Number of Singleton Blocks in Set Partition

I'm interested in some general information on the following question: Consider the collection of partitions of an $n$-set into $m$ blocks as a uniform probability space. Let $X$ be the random ...
23
votes
0answers
478 views

On the number of complete and gap-free compositions

This is a longish post about something that has been haunting me for a while about a kind of restricted composition, namely gap-free and complete compositions. First, I will define the terms that are ...
1
vote
0answers
35 views

What is the big-$\mathcal{O}$ bound for the sum of function applied to the partitions of a set?

Consider a set $A$ that is partitioned into $n$ subsets $A_1 | A_2 | ... | A_n$ and a function $f \in \mathcal{O}(g)$. Question: what is the tightest bound I can establish for $\sum_{i=1}^n ...
4
votes
1answer
1k views

Hardy Ramanujan Asymptotic Formula for the Partition Number

I am needing to use the asymptotic formula for the partition number, $p(n)$ (see here for details about partitions). The asymptotic formula always seems to be written as, $ p(n) \sim ...
4
votes
2answers
230 views

Asymptotics for partitions of $n$ with largest part at most $k$ (or into at most $k$ parts)

Let $\bar p_k(n)$ be the number of partitions of $n$ with largest part at most $k$ (equivalently, into at most $k$ parts). Is there an elementary formula for the asymptotic behavior of $\bar p_k(n)$ ...