Tagged Questions

0answers
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asymptotic approximation for number of partitions of integer that do contain 1 nor 2

Hardy and Ramanujan provided a famous asymptotic approximation to $P(n)$ the number of partitions of an integer $n$ when $n$ gets large. I wonder if there is an asymptotic approximation to ...
0answers
61 views

OEIS sequence A086449

OEIS sequence A086449 http://oeis.org/A086449 is defined by: $a(0)=1$, $a(2n+1)=a(n)$, $a(2n) = a(n)+a(n-1)+\ldots+a(n-2^m)+\ldots$ $= a(n)+\sum_{i=0}^{\lfloor\lg n\rfloor}a(n-2^i)$ One can show ...
1answer
48 views

Integer Partition into Powers

Is there any way to count the number of integer partitions of a number N into powers of two such that each size is repeated a power of two times? Ok so the recurrence can be expressed by: $a(0)=1$, ...
0answers
57 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 ...
0answers
55 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 ...
1answer
80 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 ...
0answers
49 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 ...
0answers
739 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 ...
0answers
35 views

2answers
238 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 $\bar p_k(n)$ ...