## co.combinatorics – Densities of the polynomial reciprocals in binary power series

Define density $$(f (z))$$ of a power series $$f (z) = f_0 + f_1z + f_2z ^ 2 + …$$ in the binary power series ring $$F_2 ((z))$$ as the natural density of the whole $$E_f: =$$ {$$i: f_i = 1$$}

$$D: =$$ {$$(f -1): 0$$ in $$E_f$$} is the set of densities of the reciprocals of the polynomials with a constant nonzero term.

I think every arithmetic sequence has the form $$E_ {f <-1}$$ where $$f (z)$$ is a polynomial with $$f (0) = 1$$

1) Any idea about whether $$D$$ Should it be dense anywhere in (0,1)?

2) Any idea of ​​the upper limits for $$D$$ ? inf$$(D) = 0$$

I remember that part of this arose in the "Binary Power Reciprocals" series by Cooper and Bryant led by questions about the parity of the partition function and, among other things, the densities of the polynomial reciprocals.

## co.combinatorics – The best strategy for a combinatorial game

Consider the following scenario. We have 20 balls and 100 boxes. We put the 20 balls in the boxes, and each box can contain a maximum of one ball.

Now suppose we have 5 opportunities to choose 20 out of 100 boxes. Let's say that every time we choose 20 boxes, they form a group of boxes. Then, in the end, we can choose 5 groups (each consists of 20 boxes).

Each group, of course, can contain 0 to 20 balls. In the end, we get a score, which is the maximum number of balls in any of the groups we choose.
Our goal is to maximize our score.

A bad strategy is to randomly choose the groups, in which case we will probably get a score of 0. A smarter way is to use the locker principle and choose 5 disjoint groups, in which case the locker principle tells us that at least one of The groups contain 20/5 = 4 balls. Therefore, our minimum score becomes 4.

In the general case, where we have $$N$$ boxes and $$k$$ balls, and they are given $$N / k$$ possibilities to choose size groups $$k$$, the strategy I described above seems to be optimal, which guarantees a score of $$k ^ 2 / N$$. On the other hand, if they give us $$O (2 ^ N)$$ possibilities of choosing groups, we can guarantee a maximum score of $$k$$. My question is, what is the optimal strategy to maximize our score when we are only given a polynomial Number of groups Can we do better than a guaranteed score of $$k ^ 2 / N$$?

## co.combinatorics – code to shuffle and fill

I am currently researching Zeta Multiple Values ​​(MZV), and I often need to shuffle or enter two words. Therefore, I am thinking of automating the two operations by computer code. Are there already written codes or applications or do MZV researchers write the code for themselves? If it is the latter, what language would you recommend to write that code? Mathematica, C ++, Python?

## co.combinatorics – Request the order of symbols from a combined formula

I want to create a sequence pattern using the combination formula:

nCr = n! / r! * (n – r)!

For example :
I have 5 blank zeros:
00000

I want to know how many patterns are formed if you fill in with 3 symbols (in this case I am using "1" as a symbol)

Using combined formula:

5C3 = 5! / 3! * (5 – 3)! = 10 patterns

These are the results:

to. 00111,
second. 01011,
do. 01101,
re. 01110,
my. 10011,
F. 10101,
Sun. 10110,
h. 11001,
I. 11010,
j. 11100

As we can see in the identification "a", the symbol "1" from left to right located in the third, fourth and fifth place:
00111

ID "b" the symbol "1" located in the second, fourth and fifth place:
01011, etc.

The question is:

• Is there a formula to know the order of the symbols?
• In which ID do the patterns take the following sequence:
• The symbols are in position: 2,3,4
• The symbols are in position: 1,2,4, etc.

Thank you

## co.combinatorics – Extension of Dinitz conjecture to rectangles

The Dinitz Conjecture, which Galvin later demonstrated in a more general way, stated that given a $$n times n$$ matrix, its elements could be filled exactly as a Latin square, where the elements in each row / column are chosen from a set (list) of $$m ge n$$ symbols

Is an extension of this possible for partial Latin rectangles? It is given a size rectangle. $$m times n$$ and symbols $$ge max (m, n)$$Is it possible to fill a maximum of $$min (m, n)$$ symbols in each row and / or column so that the symbols in each row, column and diagonal (places $$a_ {ij}$$ with $$j = i$$, where $$i, j$$ denote row and column indexes and $$to$$ denotes the symbol) are they different?

## co.combinatorics – Color chart of the tripartite chart

Leave $$G$$ be a tripartite chart with partite sets $$A, B, C$$. The graphics $$A cup B$$, $$B cup C$$ Y $$C cup A$$ They are each bipartite. Let the maximum degree of the graph be $$Delta$$.

Now, we know that the conjecture of the color of the border of the List is true for bipartite graphics, that is, the possibility of choosing the border is the same as the chromatic number of the border or the chromatic index for bipartite graphics. Now, first we list the color of the border of the graph $$A cup B$$, then color the chart on the edge $$B cup C$$ and finally the remaining graph $$C cup A$$. So this should give a list of chart colors $$G$$ law. The lists while the graphics are colored. $$A cup B$$, $$B cup C$$ Y $$C cup A$$ they are actually length lists $$chi & # 39; (G)$$, where $$chi & # 39;$$ is the color index Where is the flaw in this argument? Thanks in advance.

## co.combinatorics – Problem of two pairs, two sums of dice

Then there are four people, A, B, C and D.

Each rolls a 40-sided dice, one by one. If they throw a number that another person has previously thrown, they roll again. At the end of the rolls, everyone must have a unique number between 1-40.

The highest number and the lowest number of the four are paired and added. The two "middle" numbers also add up. The highest added pair earns the difference of the two sums.

For example, if they throw 1,2,3,5, then pair1 (1 + 5) – pair2 (2 + 3) = \$ 1 for pair1.

If you had to roll first, what number would you expect to get to maximize your pair's earnings? (could end in pair1 or pair2, unless you have taken 40/1)

## co.combinatorics – chromatic class of order graphics \$ n \$

Leave $$mathcal {G} (n)$$ be the isomorphism class of simple order graphics $$n$$. We say two graphics in $$mathcal {G} (n)$$ they are chromatic equivalents if their chromatic polynomials have an equal linear coefficient. The resulting equivalence classes $$chi (n)$$ under this relation they are called the chromatic class of order $$n$$.

I have the following questions :

1. What is the cardinality of $$chi (n)$$?

2. What are some references to study on $$chi (n)$$?

Thank you.

## co.combinatorics – Asymptotic estimation of cardinality of \$ H_ {2n} = { vec {a} in [-N,N]^ 2n: sum_ {i = 1} ^ {2n} a_i = 0 } \$

Suppose $$a_i sim mathcal {U} ((- N, N))$$ where $$(- N, N) subset mathbb {Z}$$. Then we can define:

$$begin {equation} S_n = sum_ {i = 1} ^ n a_i tag {1} end {equation}$$

Now to estimate $$lvert H_ {2n} rvert$$ We can try to find an asymptotic estimate of:

$$begin {equation} P (S_ {2n} = 0) tag {2} end {equation}$$

By decomposition $$S_n$$ in positive and negative parts:

$$begin {equation} S_n = S_n ^ + + S_n ^ – tag {3} end {equation}$$

where $$S_n ^ +$$ define the sum of positive terms and $$S_n ^ –$$ Define the sum of negative terms. I reasoned that the average positive and negative step length should be approximately $$Delta = frac {N} {2}$$ when $$n$$ It's big so:

$$begin {equation} P (S_ {2n} = 0) sim frac {1} {2 ^ {2n}} {2n choose n} sim frac {1} {2 ^ {2n}} frac { sqrt {4 pi n} ( frac {2n} {e}) {2n}} {2 pi n ( frac {n} {e}) ^ {2n}} sim frac {1} { sqrt { n}} tag {4} end {equation}$$

This would imply that:

$$begin {equation} lvert H_ {2n} rvert sim frac {(2N + 1) ^ {2n}} { sqrt {n}} tag {5} end {equation}$$

but I must admit that my reasoning was not very rigorous here. Could there be a rigorous estimate of $$lvert H_ {2n} rvert$$ using a probabilistic method?

## co.combinatorics – Featured examples of \$ q \$ – analogues without known cyclic screening

The phenomenon of cyclic screening is summed up very well in the following AMS notices "What is …?" Article: https://www.ams.org/notices/201402/rnoti-p169.pdf.

In that article, Reiner, Stanton and White explain some desiderata (conditions (i) – (vi)) for "very nice" $$q$$– analogues $$X (q)$$ of cardinalities $$# X$$ of combinatorial sets.

Question: What are some "very nice" $$q$$– analogues (especially those with simple product formulas) for which there is no known cyclic action that results in a CSP?