All Questions
Tagged with enumerative-combinatorics generating-functions
9 questions with no upvoted or accepted answers
4
votes
0
answers
211
views
Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions
Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
- 261
3
votes
0
answers
222
views
Number of partitions of set restricted by sum of square of part size
Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
- 405
3
votes
0
answers
137
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Positivity of sequences
Totally positive sequences $\lbrace a_n\rbrace_{n\in\mathbb{Z}}$ are defined as those such that the Töplitz matrix $A_{ij}=a_{i-j}$ is totally positive (all its minors are non-negative). An ...
2
votes
0
answers
178
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Which combinatorial class do noncrossing partitions belong to?
Let $n$ be a nonnegative integer. The set $\lbrace 1,2,\ldots, n\rbrace$ is partitioned into blocks, with $p\left(n\right)$ possibilities (e.g., for permutations $p\left(n\right)=n!).$ For each block ...
- 329
2
votes
0
answers
55
views
Compact expression for triples of subsets with total sum zero
I am looking whether there is any compact way to write the following: Suppose we have an abelian group $G$. For a subset $A\subset G$ let $S_A$ be the sum of its elements. I want to find the number of ...
- 847
2
votes
0
answers
119
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Counting the number of simple labelled bipartite graphs 𝐺𝑛,𝑚 with 𝑘 edges such that 𝑑1 vertices have degree 1
I have tried to count the number of simple labelled bipartite graphs $G_{n,m}$ with $k$ edges such that $d_1$ vertices have degree 1.
Has this problem been studied?
So far the only related paper I ...
- 21
2
votes
0
answers
66
views
Annihilator of the of the generating function not holonomic
The following is a generating function in $x,h$ with infinite parameters
$q_1,q_2\ldots,$ and $w_1, w_2,\ldots$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...
- 685
2
votes
0
answers
187
views
Direct proof that a certain generating function is D-finite
Consider the set $T^{2,3}_n$ of all non-planar rooted trees with $n$ leaves labelled by $1,2,\ldots,n$ where each internal vertex can have two or three children. If we think of the binary / ternary ...
- 16.9k
1
vote
0
answers
134
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Counting unions of unlabelled connected graphs
My question can be stated as follows: let $X$ be a hereditary family of unlabelled graphs closed under disjoint unions. Suppose we know, for each $n$, the number $c_n$ of connected graphs in X on $n$ ...
- 183