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**6**

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### 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**

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728 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**

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98 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**

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96 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**

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98 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**

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44 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**

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130 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**

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122 views

### Counting number of ways to place bags and marbles inside bags so all bags contain an odd number of marbles

There was a famous trick question posed once in math.stackexchange here. The question can be loosely translated to "Is it possible to place nine marbles into four bags so that each bag has an odd ...

**1**

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139 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**

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136 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**

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254 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 ...

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264 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 ...