infinite combinatorics – Can every number be realised as the chromatic number of a countable hypergraph?

If $H=(V,E)$ is a hypergraph and $kappa$ is a cardinal,we say a map $c:Vtokappa$ is a coloring if the restriction $crestriction_e$ of $c$ to $e$ is non-constant whenever $ein E$ and $|e|>1$. The smallest cardinal such that there is a coloring from $V$ to that cardinal is denoted by $chi(H)$.

By $(omega)^omega$ we denote the set of infinite subsets of $omega$.

Given $ninomega,n>1$ is there $Esubseteq(omega)^omega$ such that $chi(omega,E)=n$?

combinatorics – Floating-point oblivious way to compute multiset numbers

I have to compute $R = left(!!{n + 1choose k}!!right)$, which happens to be:

$$R = left(!!{n+1choose k }!!right) = frac{(n + k)!}{n!k!} = frac{(n+1)(n+2)cdots(n+k)}{1cdot 2 cdots k}$$.

The problem is that, if you compute the numerator and denominator separatedly, you will run very soon into overflow territory, so a better way to calculate this division, since both parts of the fraction have same number of componentes, is by: $$R = left(frac{n+1}{1}right)left(frac{n+2}{2}right)cdotsleft(frac{n+k}{k}right) = prod_{i=1}^kfrac{n+i}{i} = prod_{i=1}^kleft(frac{n}{i} + 1right)$$

The problem of this approach is that you have to apply floating point arithmetics to do the divisions, which are way more inneficient than using simple integral arithmetics.

But, knowing that the final result would be integral, is there any(1) way to split the big fraction into a sequence of products of fractions so that each fraction has a numerator that is a multiple of its denominator? The idea is to remove any floating point operation to calculate $R$.

Has this formula any property that guarantees that such sequence exists?

(1) Ok, maybe not “any” way would be valid. If the process to avoid the floating point arithmetics would cost more than the floating point arithmetics itself, it won’t be worthy.

additive combinatorics – What is the importance of “small doubling” in the theory of approximate groups?

One question I have is “why are approximate groups important?”. If the small doubling constant is $1$ then it’s definitely a group. If I read Green’s note correctly. (1, 2)

To be more specific let’s look at Freiman’s theorem.

Thm Let $G$ be a group and $A subset G$ be a finite subset such that $|A^2| < frac{3}{2}|A|$. There exists a subgroup $H$ with $|H| = |A^2|$ such that for every $a in A$, we have $A subset aH = Ha$.

One motivation I could see for approximate groups is that the objects we are dealing with are not quite perfectly symmetric. Perhaps the object is not quite a perfect circle, so that when we rotate it doesn’t quite map to itself $A cap R_theta A subset A$. This might have a name in the literature already. Such a shape might appear in the Number Theory or Fourier Series or something.

So why are theorems like this important? Or why can we already look at this as objects of pure study? Also what’s so special about the fraction $frac{3}{2}$ that is making the proof easier?

The lemma in the textbook doesn’t look any better. (Book)

Lemma Let $G$ be a group and let $A subset G$ be a finite subset such that $|A^2| < frac{3}{2}|A|$ then $H = A^{-1}A$ is a subgroup of $G$. Moreover $H = AA^{-1}$ and $|H| < 2|A|$.

So how “close” are we to proving the theorem here?

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co.combinatorics – Combinatorics and geometry underlying a refined Pascal matrix/Newton identities

The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for transforming between complete homogeneous symmetric polynomials/functions and elementary symmetric polynomials/functions. Certain Koszul duals are related to this.

The algebraic combinatorics of the complementary reciprocal of a Taylor series/e.g.f. is governed by the antipode/refined Euler characteristic classes of the permutahedra or, equivalently, by surjective mappings, so I have an indirect geometric combinatorial interpretation of ‘scaled’ versions of the Newton identities, but I’m looking for more direct interpretations.

What combinatoric/geometric structures are enumerated by the integer coefficients of these partition polynomials for conversion of an o.g.f. into a reciprocal o.g.f.?

combinatorics – Existence of a special in a special DAG

Suppose we’re given a Directed Acyclic Graph $G$ with $L$ leaves such that every non-leaf vertex has out-degree $2$. Let $r(v)$ the number of leaves reachable starting at vertex $v$. $G$ is such that it has a root vertex $v_0$ with $r(v_0) = L$.

How does one prove that given these conditions $G$ also has a vertex $v$ such that $dfrac{L}{3}leq r(v)leqdfrac{2L}{3}$?

It is obvious that for all leaves $l$, $r(l) = 1$. Now as we go from the root $v_0$ towards any one leaf $l_0$ we trace out a path along which $r(cdot)$ either decreases or stays the same and overall it decreases from $L$ to $1$. This is true for all such paths. But how does one prove the existence of a vertex with $dfrac{L}{3}leq r(v)leqdfrac{2L}{3}$?

Question concerning combinatorics

How many sequences of the three characters @, $, and # of length n (being a natural number) are there in which all three characters occur at least once? With reason! (Test: For n = 3 there are 6, for n = 4 there are 36 such sequences

co.combinatorics – Uniqueness in Mare combinatorics

Let $R$ be the root system of a Weyl group $W$. Let $tilde{R}^+$ be the set of $B$-cosmall roots, i.e. the set of positive roots $alpha$ such that $ell(s_alpha)=2operatorname{ht}alpha-1$. Based on the combinatorics developped in Mare’s paper, Section 3, I have the following conjecture.

Conjecture. Let $alpha_1,ldots,alpha_r,gamma_1,ldots,gamma_rintilde{R}^+$ be such that $s_{alpha_1}cdots s_{alpha_r}=s_{gamma_1}cdots s_{gamma_r}$, $$ell(s_{alpha_1}cdots s_{alpha_r})=ell(s_{gamma_1}cdots s_{gamma_r})=ell(s_{alpha_1})+cdots+ell(s_{alpha_r})=ell(s_{gamma_1})+cdots+ell(s_{gamma_r}),,$$ and such that $alpha_1^vee+cdots+alpha_r^vee=gamma_1^vee+cdots+gamma_r^vee$. Then, we have ${alpha_1,ldots,alpha_r}={gamma_1,ldots,gamma_r}$.

I proved that this conjecture is true whenever $r=2$, $alpha_1notperpalpha_2$, $gamma_1notperpgamma_2$. While the general conjecture above might be too optimistic, I would like to see a counterexample preferably for $r=2$, but could not find one, and a counterexample for $r>2$ is also welcome. Any help is appreciated!

Motivation. Concerning the motivation for this conjecture, I would like to add that an affirmative solution to the above conjecture would have implications on numerical bounds on three point genus zero Gromov-Witten invariants. For example, in type $mathsf{A}_n$, one might conjecture that $$N_{u,v}^{w,d}leqleftlceilfrac{n}{2}rightrceil!$$ for all $u,v,winmathbb{S}_{n+1}$ and all degrees $d$, where $N_{u,v}^{w,d}$ is the three point genus zero Gromov-Witten invariant of degree $d$ passing through three Schubert cycles parametrized by $u,v,w^*$, in other words, the structure constant of (small) quantum cohomology.

combinatorics – How many combinations without any repetition of pairings?

I have a quick combinatorics question. How many combinations of k elements from a set of n can be created with the restriction that no two elements can appear in the same combination more than once? So, any pair of two elements can appear in a combination only once. Is there a good way of thinking about this?

Also, is there a method for generating a list/set of the number of combinations.

combinatorics – Permutation Word Problems

Below are some word problems regarding permutation. Can anyone provide explanations for the given answers on problems 1 & 2? As well as, check out my answer on problem 3 if it’s correct? Any help will do. Thank you!

Problem #1:
From the digits 2, 3, 4, 5, 6

a. how many numbers greater than 4,000 can be formed?

Answer is 5! No explanation provided on my sheet.

b. how many 4 digits would be even?

Answer is 5! + 3 + 24 = 147. No explanation provided on my sheet.

Problem #2:
At a dinner party 6 men and 6 women sit at a round table. In how many ways can they sit if men and women alternate?

Answer is 86, 400. No explanation provided.

Problem #3:
A 4-member Student Disciplinary Tribunal (SDT) is to be formed composing of 2 faculty and 2 student members. There are 5 faculty nominees and 6 student nominees. How many ways can the SDT be formed?

In this case, there’s no answer provided on my sheet. I have an idea how to solve this, though. I assume the solution would be:

For faculty, 5!/3!
For students, 6!/4!

5!6!/3!4! = 600 ways

combinatorics – Total number of words with exactly one character larger than preceding character

I have to count the total number of words which satisfies the following property-

  1. Has a length $n$ (where $1le nle26$)
  2. Contains only unique lowercase characters
  3. Exactly one character is lexicographicaly greater than the preceding character.

Example, for $n=2$, the number of such words is $325 (ab,ac,ad,dots,bc,bd,be,dots,yz)$

I have got the formula total words $= _{26}C_{n}cdot(2^n-(n+1))$ by some observation but can’t justify it.