Unanswered Questions
3,131 questions with no upvoted or accepted answers
1
vote
0
answers
13
views
Can P-recursive functions assume only prime values?
A function $f\colon \{0,1,\dots\}\to \mathbb{R}$ is P-recursive if
it satisfies a recurrence $$
P_d(n)f(n+d)+P_{d-1}(n)f(n+d-1)+\cdots+P_0(n)f(n)=0,\ n\geq 0, $$
where each $P_i(n)\in \mathbb{R}[n]$ ...
- 49.9k
4
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0
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46
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Why is $ULU=NU$ (a refinement of $|N|=q^{n^2-n}$)?
Let $G=GL_n(\mathbb{F}_q)$, $U$, $L$, $N$ the subsets of upper-triangular unipotent, lower-triangular unipotent, all unipotent matrices respectively. Then $ULU=NU$ means that for any $g\in G$ the ...
- 3,632
2
votes
0
answers
57
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How many Tverberg partition are in cloud of points?
Tverberg's Theorem: A collection of $(d+1)(r-1) +1$ points in $\mathbb{R}^d$ can always be partitioned into $r$ parts whose convex hulls intersect.
For example, $d=2$, $r=3$, 7 points:
Let $p_1, p_2,...
- 179
0
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51
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Closed form for the A357990 using A329369 and generalised A373183
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor, \\
\ell(0) = -1
$$
Let
$$
f(n) = \ell(n) - \ell(n-2^{\ell(n)}) - 1
$$
Here $f(n)$ is A290255.
Let $A(n,k)$ be a square array such that
$$
A(n,k)...
- 4,024
0
votes
0
answers
46
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Proving we can minimize the number of crossings by having a planar embedding of $K_{2,2}$ encircle another out of any 2 such embeddings
Say that we draw a graph in the following way: we first draw $n$ planar embeddings of $K_{2,2}$ (that is, we first draw $n$ quadrilaterals) such there are no edges which cross. Then for each of the $...
- 1
0
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42
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Tamari lattice and bicategory coherence
Reference links:
Tamari lattice (Wikipedia): https://en.wikipedia.org/wiki/Tamari_lattice
Associahedra: https://en.wikipedia.org/wiki/Tamari_lattice#/media/File:Tamari_lattice.svg
The Tamari lattice ...
- 62.3k
1
vote
0
answers
82
views
Evaluating the difference of weighted binomial coefficients
I encountered the following type of sum:
$$
\begin{align}
\left[
\sum_{k=1}^{t}\binom{k+i-2}{i-1}\binom{t-k+l_1-i}{l_1-i}\sum_{s=k}^{t}\binom{t-s+l_2-j+1}{l_2-j+1}\binom{s+j-3}{j-2}
\right] \tag{1} \\
...
- 11
2
votes
0
answers
100
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What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
- 35
0
votes
0
answers
42
views
Possible determinants of 01-matrices with at most three 1s in each row, column
As a function of $n$, what is the set of possible determinants of $n \times n$ matrices whose elements are 0s and 1s and have at most three 1s in each row and column?
I really enjoyed the problem ...
- 429
0
votes
0
answers
67
views
Generating function for dimensions of the space of polynomials fixed by a single permutation
Consider the space of polynomials with complex coefficients $\mathbb{C}[x_1,x_2,\dots,x_n]$ and let $\sigma$ be a permutation of $\{1,2,\cdots, n\}$ that acts on
this space via $\sigma(x_i)=x_{\sigma(...
- 141
6
votes
0
answers
104
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Eulerian posets and order complexes
To every poset $P$ it is possible to associate its order complex $\Delta(P)$. The faces of $\Delta(P)$ correspond to chains of elements in $P$. An Eulerian poset is a graded poset such that all of its ...
- 1,949
0
votes
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57
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A weakening of the definition of positive roots for a root system
Given a root system $\Delta$ a choice of positive/negative roots is a decomposition of the elements of $\Delta$ into two subsets $\Delta^+$ and $\Delta^-$ satisfying
$$\Delta^+ = - \Delta^-\tag{$*$}\...
- 12k
5
votes
0
answers
620
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+100
For all $n\in \mathbb{N}$, How to find $\min\{ m+k\}$ such that $ \binom{m}{k}=n$?
I asked this question on MSE here.
Most numbers in pascal triangle appear only once ( excluding the duplicates in the same row of the Pascal's triangle) but certain numbers appear multiple times. ...
- 421
1
vote
0
answers
155
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Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 12.3k
0
votes
0
answers
53
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Arrangements of fixed $k$-polyplets in a $n\times n$ matrix
Recently, I asked a question about the number of arrangements of $k$ elements inside a $n\times n$ matrix with certain restrictions. The one I´m actually interested in for this question is in its 2. ...
- 12k