5 votes
Accepted

Permutation graph with insert-and-shift

Diameter equals $n-1$. For an upper bound, we should get from every permutation an identical one by at most $n-1$ such transformations. Just place 1 to its place (one transformation is always enough), ...
Fedor Petrov's user avatar
5 votes

Number of regions created by r hyper-planes in n-dimensional space

One proof is in my book Enumerative Combinatorics, vol. 1, second ed., Proposition 3.11.8. The first proof is due to L. Schläfli, written in 1850-52 but not published until 1901 in Neue allgemeinen ...
Richard Stanley's user avatar
4 votes

Order on Euclidean space in which a finite poset embeds

If $(P,\le)$ is a poset, the least $n$ such that $(P,\le)$ embeds into a product of $n$ total orders (or equivalently, such that $\le$ is the intersection of $n$ total orders on $P$) is known as the ...
Emil Jeřábek's user avatar
4 votes

A question related to "Locally Sidorenko" type problem

If I understand your question correctly, you are asking if there is a sparse graph counting lemma assuming only that the graphon is within $o(p)$ to the density $p$ in cut norm; the Lovasz result ...
Terry Tao's user avatar
  • 108k
4 votes

What is this Ramsey problem?

A good reference is Radziszowski's article Small Ramsey Numbers, which gets updated as new results are proven. In particular, this refers to basically all known Ramsey style results. The ones you're ...
David White's user avatar
  • 29.4k
3 votes
Accepted

Ask for a generating function or an explicit expression of a triangle of positive integers

The generating function: $${\cal C}(x,y) = \sum_{n,k\geq 0} C_{n,k} x^n y^{2k}$$ has the following explicit form: $${\cal C}(x,y) = \frac{\arctan(y)}{y(1-x(1+y^2))}.$$ For "one more problem",...
Max Alekseyev's user avatar
3 votes

How to efficiently sample uniformly from the set of $p$-partitions of an $n$-set?

I have provided some Python code for a quick and dirty implementation of the algorithm described by Kevin in the hope that it will save someone some time. The randomPartitionsFixedP() function is the ...
yerbles's user avatar
  • 31
1 vote
Accepted

Formula for partitions of integers with no subpartition being a partition of $t$

Let $t$ be fixed. Per Answer 1, the number of 2-forcing (nonnegative) partitions equals the coefficient of $q^M$ in Gaussian binomial coefficient $\binom{N+t-1}{N}_q$. To answer Question 1.5, it is ...
Max Alekseyev's user avatar
1 vote

Homotopical Combinatorics

Back when this question was asked, several people suggested they were not sure what was meant by "homotopical combinatorics." Well, now there's a subfield known as homotopical combinatorics. ...
David White's user avatar
  • 29.4k

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