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8 questions
1
vote
0
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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
16
votes
3
answers
676
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Plane partitions with equal margins
A plane partition of $n$ is an table of integers $A=(a_{ij})$ which add up to $n$ and non-increase in rows and columns. For example,
$$A= \begin{matrix} 331 \\
32 \ \ \\
11 \ \
\end{matrix}
$$
is a ...
- 17k
7
votes
1
answer
241
views
What is the growth rate of the number of unoriented cobordism classes?
Let $\Omega_n^O$ denote the abelian group of cobordism classes of closed, unoriented manifolds of dimension $n$, and let $d(n) := \lvert\Omega_n^O\rvert$. What are the asymptotics of $d(n)$?
It's ...
- 85.6k
2
votes
0
answers
140
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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 ...
- 34.3k
1
vote
0
answers
105
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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
3
votes
1
answer
705
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bounded partitions and bounded signed partitions of integers
Define a bounded signed partition of length $m$ and of bounded height $h$ of an integer $n$ by a relation:
$$n = \pm a_{1} \pm a_{2} \pm a_{3} \pm \dots \pm a_{m}$$ where each $a_{i}$ is a integer in ...
CommunityBot
- 1
25
votes
3
answers
2k
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Asymptotic growth of a certain integer sequence
Some time ago, while putting my nose in the Sloane's Online Encyclopedia of Integer Sequences, I came over the sequence A019568 defined as follows:
$a(n):=$ the smallest positive integer $k$ such
...
- 60.5k
3
votes
0
answers
146
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Asymptotics of partitions in at most n parts, bounded by r
I posted this question on MathStackexchange (https://math.stackexchange.com/questions/639878/asymptotics-of-partitions-in-at-most-n-parts-bounded-by-r) some time ago, but it did not receive any answer,...
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