# Questions tagged [co.combinatorics]

Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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### Decomposing square of side length $n$ into $n$ squares in a certain “maximal” way

I was wondering if anything is known about this problem. We are given a square of side length $n$ and we wish to embed $n$ smaller (integer) squares inside it such that the sum of the side-lengths of ...
98 views

### A binomial convolution of Catalan numbers vs “utterly odd numbers”

An integer is called utterly odd if the terminal string of $1$’s in its binary representation has odd length. A number $2^{k+1}m+(2^k-1)$ where $m\geq0$ (every non-negative integer has this form) is ...
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### Prove that two functions are equal only when $s \equiv \pm r^{\pm 1} \pmod{q}$

Let us fix a positive integer $q$, and let us define a functions $P: \mathbb{Z}\times \mathbb{N} \to \mathbb{Z}$ as follows: $$P(s,t) := \sum_{j=1}^t \left\lfloor \frac{j (s-1) + t}{q} \right\rfloor$$...
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### Number of integral points at minimum distance from n given points in a Cartesian plane

Given n integral points in a Cartesian plane, I want to find the number of integral points which give minimum summary distance from all n points. For example- let given points be - (1,3), (2,3), (3,3),...
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### Okada-Schur functions and the Martin boundary of the Young-Fibonacci lattice

This question is related to three earlier posts addressing properties of the Young-Fibonacci lattice $\Bbb{YF}$, namely: Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality ...
352 views

### Typo in Stanley, Enumerative combinatorics II, Cor. 7.23.9?

In Stanley, EC2, we have the following statement: I think there is a typo in the first sum after "generating function", and that $[n]_q!$ should be replaced by $(1-q)(1-q^2)\dotsb (1-q^n)$, ...
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### How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?

The question is inspired by G. Rzadkowski and M. Urlinska's examples in their paper A Generalization of the Eulerian Numbers. They refer to the discussion Expressions involving Eulerian numbers of the ...
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### Small world network regime

I have recently read Watts, D., Strogatz, S., Collective dynamics of ‘small-world’ networks, Nature 393 (1998) pp. 440–442, doi:10.1038/30918, on small-world networks, and is still not very clear to ...
### Prove that a definition of $\mathcal{I}$ does not satisfy the exchange property
For a graph $G=(V,E)$ ($V$ set of vertices and $E$ set of edges ), $\mathcal{I}$ is defined as all of the subsets $E´\subseteq E$ where the components of $(V,E´)$ that are connected are simple paths. ...