co.combinatorics – inequality of distances in a graph

It's certainly obvious but I can not understand the reason behind this.

Leave $ D = (V, A) $ be a directed graphic, $ w: A to mathbb R $ be arc weights
Y $ s in V $. Denote with $ d (s, v) $ the shortest path length
since $ s $ to $ v $ in $ D $, subject to $ w $.

If there are no negative cycles in $ D $, then we have $$ forall (u, v)
in A: d (s, v) leq d (s, u) + w (u, v) iff d (s, v) – d (s, u) leq
w (u, v). $$

co.combinatorics – Partitions of an integer with polynomials

Determine the polynomial coefficients. $$ a_0 + 𝑎_1𝑥_1 + 𝑎_2𝑥_2 + 𝑎_3𝑥_3 + ⋯ + 𝑎_𝑟𝑥_𝑟 $$ that has the property that $ ~ 𝑎_𝑛 = 𝑝 ~ $ . Where $ p $ is the number of partitions of $ n $ composed of two parts, $ p_1 $ Y $ p_2 $, where $ 1 ≤ p_1 ≤ 100 ~ $ Y $ ~ 101 ≤ p_2 ≤ 200 ~ $ for $ ~ 102 ≤ n≤ 300 $. The answer is:
$$ a_r = 0 ~ (1≤r≤101) $$
$$ a_r = 100 – | r – 201 | (102≤r≤300) $$
Can someone help me interpret this statement and in the resolution?

co.combinatorics – Strong color index of some cubic graphics

Definition (somewhat informal) A strong edge $ k $ Colorant of a cubic graph (3-regular) is a suitable $ k $ Color your edges so that any edge next to the four adjacent edges is colored with 5 colors. the strong color index $ chi_S (G) $ of a cubic graph $ G $ It is the smallest number $ k $ such that $ G $ it has a strong advantage $ k $ Colorant.

Andersen, in [1], showed that if $ G $ is a sub-cubic graph (a graph of maximum grade 3), then $ chi_S (G) le 10 $. In the same document, he proposed the following:

Guess [Andersen, 1992] There is a constant $ g $ such that if a cubic graph $ G $ is such that the circumference $ gamma (G) ge g $, so $ chi_S (G) = 5 $.

This conjecture is highly significant, since the truth would imply the truth of several notorious graphical theoretical conjectures for all graphs (cubic) of sufficiently large circumference.

As some background information for our future question, some computer investigations (still very incomplete) seem to indicate that if $ G $ has no bridges (and at least 4 circumferences, although we are not sure that this is really necessary), then $ chi_S (G) le 8 $, and also, if $ gamma (G) ge 5 $ so $ chi_S (G) le 7 $, what if $ gamma (G) ge 9 $ so $ chi_S (G) le 6 $. Finally, we have verified that the (3,17) -box listed in [2] does not have a strong edge 5 to color, and that the (3,18) -box in [2] It has a strong edge 5 to color. We are currently trying to establish the strong color index of several girth graphs of more than 9 listed in [2], and we are also trying to establish if the (3,19) -box in [2] It has a strong edge 5 to color. And we should probably see many more graphics with a small circumference. We will update this publication as this information is verified or refuted. Our calculations are oriented at this time towards graphs with a circumference greater than 4. We need to do much more in the graphs with the circumference 3 and 4 and we recognize that this is missing. There is only so much computing time available … however, we believe there is a firm basis for our main question (question 1) ahead.

Before asking our question, we need a definition.

Definition Leave a $ n $-prismatic graph is a cubic graph obtained by joining two separate circuits of order $ n $ with a perfect combination.

Our first question is then:

Question 1 [main question] Leave $ G $ be prismatic Then the strong chromatic index of $ G $ is at most 8. In addition, if the circumference of $ G $ is greater than 4, then the strong chromatic index of $ G $ is at most 7. As always in our messages, try to provide a counterexample.

The nature of a possible proof of this is almost necessarily algorithmic. An inductive test of the more general claim that the graphics with no girth gap at least 4 have a strong chromatic index at most 8 seems a bit out of reach at this time. In fact, when working with general graphs of circumference 4 and finding subgraphs that are strong 8-critical, we find more than a thousand that are not isomorphic, and they are quite large. Of course, there are some that are small and that happen a lot more often.

In [1] A linear time algorithm is provided to find a strong coloration with a maximum of 10 colors. We would also like to know if:

Question 2 Is there a fast (linear time) and simple algorithm to find a strong edge coloration 8 of a cubic graph with no bridges?

[1] Andersen, L. D., The strong color index of a cubic graph is at most 10, Discrete mathematics, 108 (1992) 231-252

[2] Royle, G. Cubic Cages, staffhome.ecm.uwa.edu.au/~00013890/remote/cages/index.html#data

co.combinatorics – Regular graph highly asymmetric

Leave $ G $ be a simple graphic connected regularly in $ n $ vertices with chromatic number $ chi $ and maximum degree $ Delta $. Then, it is implied that $ G $ is $ chi $-part Suppose we remove one of the sets of part vertices. So, what would be the maximum degree of the induced subgraph formed by the remaining vertices?

I can say with some confidence that the induced subgraph would have a maximum degree of $ chi-2 $(since the remaining partial sets must be connected to each other, otherwise the graphic would be disconnected). Also, if the graph is transitive from vertex, I think that the maximum degree of the induced subgraph would be $ Delta-1 $. Any advice and counterexample in this case? Thanks in advance.

Co.combinatorics – Around $ K $ – Tableaux increase correction

Leave $ T $ Be a standard young picture in $[n]$. Denote the RSK algorithm $ text {RSK} (w) = (P (T), Q (T)) $ for $ w in mathfrak {S} _n $, where $ P (T) $ This is the Schencted insertion boxes.

by $ 1 leq i leq j leq n $. Leave $ T_ {[i,j]$ be the SYT bias by restricting $ T $ to the segment $[i,j]$. For a skewed form $ Y $, define the rectification of Y, $ text {Rect} (Y) $ be applying jeu de taquin in $ Y $ To obtain a standard form. See section 2.1 of this document, in which $ text {Rect} $ it is denoted as $ text {std} $.

It is well known that

by $ w in mathfrak {S} _n $, $ T in text {SYT} _n $. Yes $ P (w) = T $, so
$$ text {Rect} (T_ {[i,j]}) = P (w_ {[i,j]}) $$
for all $[i,j] subseteq [n]$, where $ w_ {[i,j]$ It means restricting the permutation to the subalphabet $[i,j]$, e.g. $ 126534_ {[2,5]} = $ 2534.

My question is: is there a $ K $-the theoretical analogy of this property, in terms of Hecke's insertion, $ K $-jeu-de-taquin de tableaux growing?

Specifically. We define $ K $-rectification replacing jdt with $ K $-jdt, and denotes $ K $$ P (w) $ The Hecke insertion box for the word. $ w $.

Leave $ T $ be a growing picture (of the alphabet $[n]$) Y $ Y = T_ {[1,i]$ such that $ Y $ it's a SYT and $ 1 cdots i notin T backslash Y $ (so there is no ambiguity).
Then it is always true that:
$$ K text {-Rect} (T backslash Y) = K text {-} P (w_ {[i+1,n]}) $$?
where $ w $ is the row reading word of $ T $, or even all the words that $ K text {-} P (w) = T $.

For example, Let $ T = begin {matrix} 1 & 2 & 4 \ 3 & 5 & 6 \ 4 & 6 & 9 end {matrix}, Y = begin {matrix} 1 & 2 \ 3 & end { matrix} $. We have
$$ K text {-Rect} left ( begin {matrix} * & * & 4 \ * & 5 & 6 \ 4 & 6 & 9 end {matrix} right) = begin {matrix} 4 & 5 & 6 \ 5 & 9 & \ 6 && fin {matriz} $$
The row reading the word of $ T $ is $ w = mathfrak {row} (T) = 469356124 $Y
$$ K text {-} P (w_ {[4,9]}) = K text {-} P (469564) = begin {matrix} 4 & 5 & 6 \ 5 & 9 & \ 6 && end {matrix} $$

Thank you!

co.combinatorics – Probability that a random collection of subsets is a cover

Consider the whole $[n]= {1,2, ldots, n } $. Suppose that for each set $ A subseteq [n]$ I have a $ p_A in [0,1]$. Now I create a random collection $ mathcal {W} subseteq mathcal {P} ([n]$ of subsets of $[n]$ including each $ A $ with probability $ p_A $independently What is the probability that your union will cover? $[n]$, that's that
$$ bigcup_ {W in mathcal {W}} W = [n]$$

This seems to be a problem that has been absolutely considered before, it is easy to say and it seems natural, and the answer should be "simply" a properly symmetric multivariate polynomial. Unfortunately, I can not solve it on my own, and my google-fu has not helped me either.

co.combinatorics – Limits on the chromatic number when the maximum degree is large

For a regular graph with $ n $ vertices and maximum degree $ Delta $, it is easy to see that the chromatic number, $ chi le frac {n} {2} $ Yes $ frac {n} {2} le Delta lt n-1 $(from a regular graph in $ n $ vertices with maximum degree $ n-2 $ is the complete graph with a factor eliminated, which will have each vertex not adjacent to another single vertex, which could be given the same color, using the lemma of the handshaking we obtain the chromatic number of said graph. $ frac {n} {2} $)

How could this fact be applied to match the chromatic number of any non-regular graph with a large maximum degree? Does this fact have a well-known name, such as Reed's theorem or Brooks's theorem? Thanks in advance.

co.combinatorics – Number of Dyck routes up to stable equivalence

The acyclic (connected) Nakayama algebras can be identified with the Dyck routes through their upper limit Auslander-Reiten alches.

Now two Nakayama algebras $ A $ Y $ B $ It must be stable equivalent in case of eliminating the projective vertices in its routes Dyck gives the same result until the order. (I hope this is correct, since my experience with stable equivalences is very limited and I may have a thinking error).

Note that after elimination we obtain a disjoint union of the Dyck routes.

Dyck roads $ D $ they are always a sequence of irreducible Dyck routes (that do not touch the x-axis, except at the beginning and at the end).

Then, call two stable equivalent Dyck routes in case your sequence of irreducible Dyck routes is the same as that of a multiple set. In case you do not have any thinking errors, this should describe when the acyclic Nakayama algebras are stable equivalents using their associated Dyck paths.

For example, the Nakayama algebras with the Kupisch series. [3,2,2,1] (= UDUUDD) and [2,3,2,1] (= UUDDUD) must be stable equivalent since the resulting stable Dyck route is in both cases the disjoint junction of the Dyck UD route and the Dyck route consisting of only one point.

Question: Is there an explicit formula for the number of Dyck routes up to stable equivalence?

Here is the generating series:

Generating series of the irreducible paths of Dyck is $ B (x) = x frac {1- sqrt {1-4x}} {2x} $ and when composing this with the Multiset generation series, the number of Dyck routes up to the stable equivalence must have the following generation series:

$ e ^ { sum limits_ {k = 1} ^ { infty} { frac {B (z ^ k)} {k}}} $ with $ B (x) = x frac {1- sqrt {1-4x}} {2x} $. I wonder if you can get an explicit formula for the sequence of the generating series.

co.combinatorics – Turn-based group of a toroidal cube

To consider $[n]^ d $ — a $ d $Three-dimensional toroidal cube with lateral length $ n $ divided in $ n ^ d $ cubes unit. Define $ k $– Change as a next type of permutation in unit cubes: choose $ S subset [n]$ with $ | S | = k $, $ i in S $, and assign a number $ y_j in [n]$ to each $ j no in S $. For a cube unit. $ (x_1, ldots, x_d) $, we change $ x_i $ to $ x_i + 1 $ with loop over the edge if $ x_j = y_j $ for each $ j no in S $Otherwise, the cube remains in place. For example, a $ 1 $-the change in a toroidal square would be the cyclic displacement of a row or a column, a $ 2 $-the change in a 3D toroidal cube is a change of a flat layer, and so on.

Leave $ G_d (n, k) $ Be the group induced by all possible. $ k $-changes in $[n]^ d $. This question has to do with identification $ G_d (n, k) $, or at least some of its properties.

  1. In which cases can you identify $ G_d (n, k) $? What is its size, simple subgroups, composition series?
  2. Clearly, for any $ k in [d – 1]$ we have that $ G_d (n, k + 1) $ it's a subgroup $ G_d (n, k) $. Can we answer some of the previous questions about $ G_d (n, k) / G_d (n, k + 1) $? There is $ n, k, d $ such that the previous quotient is trivial?
  3. Given a permutation $ sigma in mathrm {Sym} ([n]^ d) $Can the highest be determined efficiently? $ k $ such that $ sigma in G_d (n, k) $?

Choice of $ n $ is not particularly important, for simplicity we can take $ n $ to be a prime number

co.combinatorics – Circular permutations (bracelets) with similar things (reflections are equivalent) using the enumeration of polia

The circular permutations of N objects of n1 are identical of one type, n2 are identical of another type and so on, so that n1 + n2 + n3 + ….. = N?
There is a similar question but it does not address the case in which the reflections are under the same equivalent class.$$ frac {1} {N} sum_ {d | N} phi (d) p_d ^ {N / d} $$ This is when the reflections are not the same. How does the equation change under this new restriction?

Note: I could not comment on that question because of my low reputation, so I asked this question.