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6
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0answers
203 views

Counting Selections of Entries such having an Extremal Permutation of length n^2+1

Let $S_{n^2+1}$ be permutations of length $n^2+1$. By Erdos-Szekeres Theorem. any $s \in S_{n^2+1}$ would have a monotone subsequence(increasing or decreasing)of length $n+1$. Say a permutation $s$ of ...
5
votes
0answers
712 views

On “The Average Height of Planted Plane Trees” by Knuth, de Bruijn and Rice (1972)

I am trying to derive the classic paper in the title only by elementary means (no generating functions, no complex analysis, no Fourier analysis) although with much less precision. In short, I "only" ...
3
votes
0answers
91 views

Counting Problems where Labeled is Known but Unlabeled is Not

Cayley's formula states that the number of labeled trees on $n$ vertices is $n^{n-2}$. There are many nice proofs of this compact formula. To contrast, counting unlabeled trees is considerably ...
3
votes
0answers
89 views

Counting perfect matchings with integrals

Has anyone used the Joni-Rota-Godsil integral formula (see details below) to count perfect matchings of square-grid graphs, Aztec diamond graphs, hexagon-honeycomb graphs, etc.? (Or even just to ...
3
votes
0answers
97 views

Enumerating simple algebraic groups and their irreducible representations

Motivation Everything is over an algebraically closed field. Given a faithful representation $G \to \textrm{GL}(V)$, one may try to pin down what the group $G$ exactly is (e.g., the in the case of ...
2
votes
0answers
36 views

Counting labelled graphs according to sets of size 3

In this question we are counting labelled simple graphs. No concept of isomorphism is involved. Let $G(n,e,t)$ be the number of labelled simple graphs with $n$ vertices, $e$ edges, and $t$ sets of ...
2
votes
0answers
121 views

combinatorial rectangles

Consider the set $S$ of all $m\times m$ matrices with $0-1$ entries with exactly $T$ combinatorial rectangles of all $0$s or all $1$s that partition each matrix in a non-overlapping manner. Is there ...
1
vote
0answers
115 views

Distinct determinants of circulants

If $M$ is a circulant integer matrix of size $n\times n$ whose entries are randomly chosen from $\{0,1\}$ value, how many different determinants does $M$ possibly take value in? For $n=1,2,3,4$, I ...
1
vote
0answers
126 views

Distinct Numbers

Let $x_{ij}\in\{0,1\}$ for ${i=1}$ to ${m}$ and for ${j=1}$ to $n$. How many different values does $$\prod_{i=1}^m\sum_{j=1}^nx_{ij}$$ cover? Is there an $a_{ijk}\in\Bbb R$ (there is a $a_{ijk}\neq0$ ...
1
vote
0answers
252 views

An Pure intriguing counting problem of index sets

Hi Guys. The problem here seems like a homework, but I think that it is not that easy.It comes from a theorem I recently proved.The content of the theorem is not important, the issue is that I have no ...
0
votes
0answers
262 views

Enumerating m-tuples of Integers Subject to Implication Constraints

How do I enumerate all $m$-tuples of positive integers $(a_1,...,a_m)$ subject to the following constraints? For each $i$ in \${ 1,\ldots,m }\$, there is a number $n_i \geq 0$ such that $a_i \leq ...