11 votes
Accepted

Explicit expression for recursive sums

Claim: The iterated sum $f_k(t_1,\ldots,t_k)$ counts the number of elements the interval $[\emptyset,\lambda]$ of Young's lattice, where $\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_k)$ is the ...
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9 votes
Accepted

Prove positivity of rational functions

Notice that $$F_r(z) = \frac{1}{(1-z)^{r-1}} - \sum_{k=0}^{r-1} \left(\frac{z}{1-z}\right)^k$$ and therefore for $r\geq 4$ and $n\geq 1$, we have \begin{split} [x^n]\ F_r(z) &= \binom{n+r-2}{r-2} -...
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9 votes

Explicit expression for recursive sums

$f_k(t_1,\dots,t_k)$ is counting the number of integer points in the "Pitman-Stanley polytope" $\Pi_k(t_1,\dots,t_k)$ defined here. The notation $N(\Pi_k(\mathbf{t}))$ is used in this paper, ...
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6 votes
Accepted

Find all 2-planar drawings of $K_6$ and $K_7$

The list of all good drawings of $K_6$ can be found in the doctoral thesis by Nabil H. Rafla: https://escholarship.mcgill.ca/concern/theses/x346d4920 On pages 164-165 the drawings are described by the ...
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  • 5,406
4 votes

On permanent of a square of a doubly stochastic matrix

A related question is Conjecture 17 in Minc's catalogue of open problems on permanents. He attributes it to Foregger. The conjecture is that for any positive integer $n$, there exists an integer $k=k(...
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2 votes

Bell polynomial with variables 1 and 0

From the generating function: $$\sum_{n\geq k \geq 0} B_{n,k}(x_1,\ldots,x_{n-k+1}) \frac{t^n}{n!} u^k = \exp\left( u \sum_{j=1}^\infty x_j \frac{t^j}{j!} \right)$$ it follows that $$\sum_{n\geq k \...
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1 vote

Strongly minimal covers for clique hypergraphs of graphs

I can prove the following weaker statement, which was not true in the other version: Every hypergraph stemming from the cliques of a graph has a minimal cover, where I define a cover as minimal if the ...
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  • 17.1k
1 vote
Accepted

Counting Euler circuits through labelled trees where $v_1$ and $v_2$ have distance two

Wlog we can use $v_{n-2}$ and $v_{n-1}$ instead of $v_1$ and $v_2$. Then if we let $B_n \subset T_n$ be the set of labelled trees with edges $(v_{n-2}, v_n)$ and $(v_{n-1}, v_n)$, the count for $A_n$ ...
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  • 3,982

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