co.combinatorics – Solving a recurrence relation involving binomial coefficients

This question originates from a graph neural network architecture (see 1) in which an edge-labelled graph $G=(V,E,eta)$ of size $|V|=n$ with $eta:Eto mathbb{R}^{s_0}$ is represented as a “tensor” $mathbf{A}_G^{(0)}in mathbb{R}^{ntimes ntimes s_0}$, and in which for round $t>0$, a new tensor $mathbf{A}_G^{(t)}inmathbb{R}^{ntimes ntimes s_t}$ is defined in terms of $mathbf{A}_G^{(t-1)}inmathbb{R}^{ntimes ntimes s_{t-1}}$. The dimension $s_t$ is defined as ${n + s_{t-1}-1 choose s_{t-1}-1}$ for $t>0$.

Question: Given $n$ and $a_0$ in $mathbb{N}$, define $a_t:={n+a_{t-1}choose a_{t-1}}$ for $t>0$. Is there an explicit formulation of $a_t$ in terms of $n$, $a_0$ and $t$? In other words, how to “solve” such a recurrence relation? What can one say about $a_t$ when $t$ is large?

co.combinatorics – Are complete regular linear hypergraphs on $omega$ isomorphic

If $H_i = (V_i, E_i)$ are hypergraphs for $i=1,2$ then we say they are isomorphic if there is a bijection $f: V_1 to V_2$ such that for $A subseteq V_1$ we have $$Ain E_1 text{ if and only if } f(A) in E_2.$$

We say that $H=(omega, E)$ is a complete regular linear hypergraph on $omega$ if

  1. $e, fin E implies |ecap f| = 1$ and
  2. for all $nin omega$ we have $|{ein E: n in e}| = aleph_0$.

Are there complete linear hypergraphs $H_i = (omega, E_i)$ for $i = 1,2$ such that $H_1, H_2$ are not isomorphic?

co.combinatorics – Invertibility of discrete Laplacian

In QFT and Statistical Mechanics the discrete Laplacian usually plays a key role when we want to discretize the theory. However, few books (at least to my knowledge) really work the properties of this operator in details, so I’m trying to figure out some of these properties myself.

Let $Lambda := epsilon Z^{d}/Lmathbb{Z}^{d}$ be a finite lattice where $epsilon> 0$ and $L > 1$ are integers such that $L/epsilon in mathbb{N}$ is even. An scalar field over the lattice $Lambda$ is simply a function $phi : Lambda to mathbb{C}$, so that the space of all fields is $mathbb{C}^{Lambda}$. Because the lattice is a quotient space, we’re dealing with periodic boundary conditions. Thus, we can introduce the discrete Laplacian as the linear operator $-Delta: mathbb{C}^{Lambda} to mathbb{C}^{Lambda}$ defined by:
$$(-Delta phi)(x) := frac{1}{epsilon^{2}}sum_{k=1}^{d}(2phi(x)-phi(x+epsilon e_{k})-phi(x-epsilon e_{k}))$$
with ${e_{1},…,e_{d}}$ being the canonical basis for $mathbb{R}^{d}$.
Now, let $langle phi, varphi rangle_{Lambda} := epsilon^{d}sum_{xin Lambda}overline{phi(x)}varphi(x)$ be an inner product on $mathbb{C}^{Lambda}$. If I’m not mistaken, the follwing identity holds:
begin{eqnarray}
langle phi, -Delta phirangle_{Lambda} = sum_{xin Lambda}sum_{ysim x}|phi(x)-phi(y)|^{2} = sum_{xin Lambda}sum_{ysim x}(overline{phi(x)}-overline{phi(y)})(phi(x)-phi(y)) tag{1}label{1}
end{eqnarray}

where $ysim x$ denotes that $|x-y| = 1$, where $|cdot|$ is the maximum ‘norm’ on $mathbb{Z}^{d}$.

My point is the following. We could have assumed $phi = 0$ outise $Lambda$ as a boundary condition, instead of our periodic one. In this case, I know that the discrete Laplacian is positive in the sense that:
$$langle phi, -Delta phi rangle_{Lambda} > 0 quad mbox{if} quad langle phi, phi rangle_{Lambda} > 0$$
and I’d expect the same property with periodic bondary conditions. However, because of the first equality in relation (ref{1}), it seems that if we take $phi$ to be constant everywhere, say $phi(x) = 1$ for every $x in Lambda$, it’d follow that $langle phi, -Delta phi rangle_{Lambda} = 0$ even with $langle phi, phirangle_{Lambda} > 0$. This would lead to a non-invertibility of this operator. I don’t know if this is a known fact that I just didn’t know yet or if my reasoning is not correct, but I’d appreciate any help here.

co.combinatorics – Euclidiean Algorithm – MathOverflow

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co.combinatorics – Sum over 0-1 matrices

I stumbled across the following formula when working on a research problem in theoretical computer science. I checked its correctness up to $N=5$ with a computer. I am looking for a simple proof of it, or any idea which might prove useful.

This question was first asked on Maths StackExchange two weeks ago.


Basic version

Let $mathcal M_N$ be the set of all invertible 0-1 square matrices of size $N$. More formally, one could write $mathcal M_N = {0,1}^{Ntimes N} cap GL_N(mathbb R)$.
$$
sum_{M in mathcal M_N}
frac{det(M)^2 cdot (-1)^{|M|_0 – N}}
{prod_{i=1}^NBig(sum_{j=1}^N M_{i,j}Big)prod_{j=1}^NBig(sum_{i=1}^N M_{i,j}Big)} = 1
$$

where $|M|_0 = sum_{i,j} M_{i,j}$ is the number of non-zero entry of $M$.


Weighted generalization

Note that the formula is also true when a “positive weight” is associated to every coefficient. Let $P$ be any matrix with positive coefficients. Let $P circ M$ be the elementwise product of $P$ and $M$.

Redefine $mathcal M_N$ to be the set of all 0-1 square matrices without any row/column of zeros (one can also define $mathcal M_N$ to be A227414)

$$
sum_{M in mathcal M_N}
frac{det(P circ M)^2 cdot (-1)^{|M|_0 – N}}
{prod_{i=1}^NBig(sum_{j=1}^N (P circ M)_{i,j}Big)prod_{j=1}^NBig(sum_{i=1}^N (P circ M)_{i,j}Big)} = 1
$$

Here is some python code to check (empirically) my claim (slow when $N > 4$).

from sympy import Matrix
from itertools import product
N = 2
result = 0
P = Matrix(((4,2),(13,37)))
for p in product((0,1), repeat=N**2):
  M = Matrix(p).reshape(N, N).multiply_elementwise(P)
  val = (-1) ** (sum(p)-N) * M.det() ** 2
  if val != 0:
    for i in range(N):
      val /= sum(M(:,i))
      val /= sum(M(i,:))
    print(M, val)
    result += val
print(result)

And for those of you who don’t want to run this program, here is the output.

Matrix(((0, 2), (13, 0))) 1
Matrix(((0, 2), (13, 37))) -1/75
Matrix(((4, 0), (0, 37))) 1
Matrix(((4, 0), (13, 37))) -74/425
Matrix(((4, 2), (0, 37))) -74/117
Matrix(((4, 2), (13, 0))) -13/51
Matrix(((4, 2), (13, 37))) 3721/49725
1

co.combinatorics – Powerful existence theorems with mild conditions: more recent examples

I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that a minimum regular pattern always exists. By mild conditions I mean short, easy, broad conditions. By simple conditions I mean not requiring advanced mathematical education. The conditions and the statement should be accessible to undergraduate mathematics/science students.

I am interested mostly in low-dimensional examples which allow an easy graphical representation.

I have some obvious examples in mind (given below), but they are rather classical results that were established between 1900 and 1950, roughly speaking.
I would be interested to see examples that are more recent.

Classical examples I have in mind

(1) Lemma of Sperner and Brouwer Fixed Point Theorem (for $n=2$)

(2) Lemma of Tucker and Borsuk-Ulam Theorem (for $n=2$)

(3) Ramsey’s Theorem (for the simplest case of 6 edges)

(4) Wagner’s Theorem about Planar Graphs

I would be grateful if you could point me to more recent examples.

co.combinatorics – Please, help! Is it possible to fill the 10×10 square by the numbers -1, 0, 1 so the 20 sums of the numbers in the rows and columns are all different?

Is it possible to fill the 10×10 square by the numbers -1, 0, 1 so the 20 sums of the numbers in the rows and columns are all different?

co.combinatorics – Relationship between the expected values ​​of the eigenvalues ​​of the Laplacian matrix of a graph and the eigenvalues ​​of the expected Laplacian matrix of that graph?

In particular, I am dealing with Erdős – Random Rényi 𝐺 (𝑛, 𝑝), so the expected Laplacian matrix of 𝐺 (𝑛, 𝑝) is 𝑝 (𝐽𝑛 – 𝐼𝑛), where 𝐽𝑛 and 𝐼𝑛 are one and identity matrices, respectively .

Also, if the distribution (uncertain, but could be the power law) of the eigenvalues ​​of the Laplacian matrix of the graph 𝐺 (𝑛, 𝑝) is known, then it seems to me that the expected value of the eigenvalues ​​has some formula closed depending on 𝑛 and 𝑝 in the asymptotic case.

co.combinatorics: How many ways to cover an N × N chessboard with black and white squares with some restrictions?

Suppose we have an N × N chessboard and the boxes ■, □.

We should cover the chess board with those boxes but it can't have the 2 × 2 square
□□
□□ (sorry, I don't know how to write the 2 × 2 form).
on the chess board.

Can we calculate the ways to cover the chessboard?

Further,
(1) if we connect the upper (left) boundary and the lower (right) boundary together, which means that the upper (left) and lower (right) boundary form a 2 × N (N × 2) rectangle and should not have the square 2 × 2
□□
□□
too.
Can we calculate the shapes?

(2) if we have k black boxes, with the (1) chess board, can we calculate the shapes?

co.combinatorics: ranking based on value (not position); reference request

A recent Ville Salo response on the diameter of a Cayley graph induced by bubble generators (adjacent transpositions) has inspired this variation.

Many sort algorithms are position-based: go through a list of pairs of positions in the (integer) array, check the two values, and trade or not. For many simple algorithms (especially in classification networks), the list of position pairs is fixed, while for less simple algorithms, the list can be created based on the input (namely Quicksort).

The previous question uses a limited list of pairs of positions (adjacent transpositions) to investigate. Suppose we use a limited list of values ​​instead?

For the next bit of the post, suppose we have scanned an array representing a permutation of the first n positive integers. We can look at everything we want, but our exchanges are limited to exchanging adjacent values: if k and k + 1 are integers in the matrix, we can interchange them, otherwise we will continue searching. Therefore, we never interchange the two values ​​3 and 8 in this scheme, for example.

There may be a duality between indices (position labels) and values ​​that could be exploited to answer questions like the diameter of the analog graph, but I still don't see it. The first question for me is: is there such an exploitable duality?

The second official question for this publication is: Has the idea of ​​(restricted) value-based classification been explored in the literature? If so, what literature and by what name?

Gerhard "Ordering with a different priority" Paseman, 2020.05.08.